Unsolved problems in QCD:a lattice approach

Alejandro Vaquero Aviles-Casco

Departamento de Fısica TeoricaUniversidad de Zaragoza

Contents

Agradecimientos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

I Symmetries of QCD and θ-vacuum 1

1 The strong CP problem 3

1.1 The symmetries of QCD lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Solution to the U(1)A problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The topological charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 The anomaly on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Spontaneous Symmetry Breaking 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The p.d.f. formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

The p.d.f. formalism for fermionic bilinears . . . . . . . . . . . . . . . . . . . . . 16

3 The Aoki phase 19

3.1 The Standard picture of the Aoki phase . . . . . . . . . . . . . . . . . . . . . . . 21χPT analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 The p.d.f. applied to the Aoki phase . . . . . . . . . . . . . . . . . . . . . . . . . 27The Gibbs state, or the ǫ−regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 27QCD with a Twisted Mass Term, or the p−regime . . . . . . . . . . . . . . . . . 29Quenched Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Unquenched Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Realization of symmetries in QCD 45

4.1 Parity realization in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Review of the Vafa-Witten theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 45Objections to the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46The p.d.f. approach to the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 51One steps further: Parity and Flavour conservation in QCD . . . . . . . . . . . . 54

4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 The Ising model 63

5.1 Introduction to the Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63The origins of the Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Why the Ising model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Solving the Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Definition of the Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

The cluster property for the Ising model . . . . . . . . . . . . . . . . . . . . . . . 68

Analytic solution of the 1D Ising model . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Systems with a topological θ term . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Ways to simulate complex actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Ising’s miracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Computing the order parameter under an imaginary magnetic field . . . . . . . . 79

Numerical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 The Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

An introduction to Mean-field theory . . . . . . . . . . . . . . . . . . . . . . . . . 88

Antiferromagnetic mean.field theory . . . . . . . . . . . . . . . . . . . . . . . . . 90

The phase diagram in the mean-field approximation . . . . . . . . . . . . . . . . 93

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

II Polymers and the chemical potential 97

6 QCD and the chemical potential 99

6.1 The chemical potential on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Polymeric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

The first steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Fall and rise of the MDP model . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3 Karsch and Mutter original model . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4 The IMDP algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

IMPD vs Worm Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Algorithms for the pure gauge action 121

7.1 The strong coupling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2 Measuring observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Plaquette foils and topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Numerical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3 The three dimensional Ising gauge model . . . . . . . . . . . . . . . . . . . . . . 133

7.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Further applications of polymers 137

8.1 Mixing fermions and gauge theories: QED . . . . . . . . . . . . . . . . . . . . . 137

Pairing gauge and fermionic terms . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Some ideas for configuration clustering . . . . . . . . . . . . . . . . . . . . . . . . 139

A word on Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.2 Other interesting models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

The Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Summary 147

Outlook 153

A Nother’s theorem 155

B Saddle-Point equations 157

B.1 The Hubbard-Stratonovich identity . . . . . . . . . . . . . . . . . . . . . . . . . . 157B.2 The Saddle-Point Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C Mean-field theory 161

D Solution to the saddle-point equation 167

E Ginsparg-Wilson operator spectrum 171

Agradecimientos

Esta compilacion del trabajo realizado a lo largo de cuatro anos no hubiera sido posible sin laayuda de multitud de personas, tanto en el ambito cientıfico como en lo personal. Puesto quela Tesis solo la firmo yo, voy a decicar unas paginas, en ejercicio de justicia, a todos aquellosque, de una manera u otra, la habeis hecho posible.

En primer lugar debo agradecer a mi director de tesis, Vicente, su apoyo y guıa, tanto a nivelcientıfico como personal. Siempre ha sido enormemente comprensivo con mis compromisos, y supreocupacion por mı nunca se limito a la fısica teorica. Todo esto hace que ahora lo consideremas un amigo que un colega.

Algo parecido se puede aplicar a mis otros colaboradores cercanos, como Eduardo y Giuseppe,a los que ademas debo agradecer su hospitalidad, y a Fabrizio, por su sabio consejo. Estos cuatroanos no hubieran sido lo mismo de no ser por Pablo, cuyas conversaciones, que solo de cuandoen cuando versaban sobre fısica y a menudo me hacıan mirar al mundo desde otro punto devista, y su companıa en el despacho han hecho que desarrolle una relacion con el que va mas allade lo profesional. Tampoco podrıa olvidarme de mis otros companeros de pasillo, en particularFernando Falceto y Jose Luis Alonso, que siempre han estado dispuestos a echarme una manocon cualquier problema que me surgiera, y me han animado a participar en proyectos nuevosque mantuvieran vivo mi interes en la investigacion; ası como de Diego Mazon y Victor Gopar,con quienes he compartido algunos buenos momentos que me han ayudado a hacer mas ligeromi trabajo. Crıticos para sobrevivir a la inmensa burocracia espanola han sido Pedro Justes,Esther Hernandez y Esther Labad. Sin ellos probablemente aun estarıa haciendo papeleo.

Fabio y Paula merecen un lugar aparte: Ademas de ofrecerme una ayuda inestimable entemas de fısica, son capaces de transformar cada escuela de fısica o cada nuevo congreso enuna fiesta. Aunque casi siempre se encontraran a varios cientos de kilometros de distancia, hanestado muy presentes desde que nos conocimos en aquella escuela de Seattle en 2007.

Al capitan Jul y a Anuchi la princesa les debo miles de horas de buenos y malos momentos.Para mı es una inmensa suerte el poder contar con vosotros. Y contigo, Julieta, que no teolvido, a pesar de la enorme distancia que nos separa. A Irenilla por saber escucharme comonadie jamas lo ha hecho nunca, una pena que te mudaras a Barcelona. Y a Carlos Sanchez,Cristina Azcona, Pepa Martınez (siempre tan dulce y tan macarra al mismo tiempo) y EnriqueBurzurı, que aunque os vea solo de cuando en cuando, siempre son agradables vuestras visitas.

No podıan faltar ente estas lıneas mis companeras de patinaje, en particular mi entrenadoraLaura, Luci y Sara, mi pareja de patinaje, con las que he compartido infinitas alegrıas, lloros ypeleas. En cuanto a mi carrera pianıstica, a Luis debo agradecerle su inagotable comprension portodos eso deberes inacabados. Y a Laura Rubio le debo mucha inspiracion, tanto para la fısicacomo para la musica. Todos ellos me han ofrecido otra vıa de escape para descansar un pocola mente cuando los problemas fısicos no encontraban solucion. Mis inseparables companerosde aventuras, Juan, Dano, Marta, Fabio y el Leas, tambien se merecen unas lıneas, por estarallı (casi) todos los domingos, dispuestos a ayudarme a desconectar machacando orcos o lo queconviniera en el momento.

Por ultimo, a mis padres, mi hermano Carlos y mi prima Ana por su apoyo incondicional yconstante.

Zaragoza, January 20, 2011

Alejandro Vaquero Aviles-Casco ha realizado su tesis gracias al programa FPU del ministeriode Educacion, a la colaboracion INFN-MEC, a la colaboracion europea STRONGnet, y al grupode Altas Energıas de la universidad de Zaragoza.

i

ii

Introduction

Quantum Chromo-Dynamics (QCD) is a challenging theory: the strong CP problem or itsbehaviour for very high densities are some of the yet-unsolved problems that worry the QCDpractitioners all along the world. Even some of its fundamental properties, like the confinementof quarks, remain unproved1.

One of the points that makes the analysis of QCD so intrincate is its non-perturbativebehaviour at low energies. Contrary to what happens in Quantum Electro-Dynamics (QED),whose coupling constant αQED is small, QCD is only accessible to perturbative techniques inthe high energy region, where its coupling constant αQCD becomes small, due to the propertyof asymptotic freedom. For low energies, αQCD takes very high values, and the perturbativeexpansion does not converge any more: we need an infinite number of Feynmann diagrams tocompute any observable. Moreover, perturbation theory is unable to give account for all thetopological properties of QCD. Since the topology plays a fundamental role in the theory, theavailability of a non-perturbative tool to study QCD is very desirable.

The most prominent tool to investigate non-perturbative theories is the lattice. Logically,in this work, which is about QCD, the lattice is the underlying basis for every development.Lattice QCD and numerical simulations usually come together: in most cases, lattice theoriesare very complex to study analitically, but surprisingly easy to put in a computer. Taking intoaccount the increasingly high computer power available nowadays, lattice QCD simulationsare becoming a powerful and accurate tool to measure the relevant quantities of the theory.Unfortunately, the lattice is far from perfect, and there are still obstacles (like the sign problem)which prevent some simulations to be performed.

The title of this thesis summarizes the aim seeked along these four years of research: totackle some of the yet-unsolved problems of QCD, to explore or create techniques to study thenature and the origin of the problem, and in some isolated cases, to find the desired solution. Asthe topics developed in this work have been the responsible of the headaches and frustrationsof many scientist during the last decades, only in very few cases a reasonable solution to theproblem exposed is found. In any case, the effort put into these pages constitute valuableinformation to those who decide to embark themselves in such a similar task.

The work exposed here can be divided in two parts. The first one (comprising chapters oneto five) is related to the chiral anomaly, the strong CP problem, and the discrete symmetries ofQCD, mainly Parity. All these topics are introduced very briefly in chapter one, to make themanuscript as self contained as possible. The second chapter is devoted to the key tool usedto study all the aforementioned phenomena: the Probability Distribution Function (p.d.f.)formalism, a well-known technique used to study the spontaneous symmetry breaking (SSB)phenomena in statistical mechanics, brought to quantum field theories. Of particular interest isthe extension of the domain of application of this technique to fermionic bilinears, well explainedat the end of the second chapter. This chapter thus establish the theoretical framework requiredto understand the analyses performed in the following pages.

The third chapter is the first one containing original contributions. It begins with an in-troduction to the Aoki phase, explaining its origin and properties from several points of view.The Aoki phase is a consequence of the explicit chiral symmetry breaking of Wilson fermions.In this phase, Parity and Flavour symmetries are spontaneously broken, and even if the Aokiphase is not physical, its existence represents an obstacle to the proof of Parity and Flavourconservation in QCD. The new research performed here is the application of the p.d.f. to thestudy the breaking of Parity and Flavour. Surprisingly we obtain unusual predictions: new

1At this moment, only numerical ‘proof’ of the confinement exists. In fact, the confinement of quarks is oneof the millenium problems, awarded with a substantial prize by the Clay Mathematics Institute.

iii

undocumented phases appear, that seem to contradict the standard picture of the Aoki phase;more exactly, the χPT and the p.d.f. predictions clash and become difficult to reconcile. SinceχPT is one of the pillars of the lattice QCD studies, it is quite important to find if one ofthe approaches is giving wrong results. The resolution to the contradiction is seeked –withoutmuch success- in a dynamical simulation of the Aoki phase without a twisted mass term, anoutstanding deed never done before, due to the small eigenvalues of the Dirac operator thatcause the Aoki phase.

The chapter four is, in some way, a continuation of chapter three. It tries to answer a questionthat arises after a careful examination of the Aoki phase: are Parity and Flavour spontaneouslybroken in QCD? The old Vafa and Witten theorems on Parity and Flavour conservation arereviewed and critiziced, and a new and original proof for massive quarks, within the frame of thep.d.f., is developed. This proof makes use of the Ginsparg-Wilson regularisation, which lacksthe problems of the Wilson fermions, despite of being capable of reproducing the anomaly on thelattice without doublers: its nice chiral properties depletes the spectrum of small eigenvalues formassive fermions and forbids the appearance of an Aoki-like Parity or Flavour breaking phase.

The fifth chapter drifts from the trend established in chapters three and four, in order todeal with the simulations of physical systems with a θ term. Although the main goal behindthis chapter is to study QCD with a θ term, it is devoted entirely to the antiferromagneticIsing model within an imaginary magnetic field, which is used as a testbed for our simulationalgorithms. The first half of the chapter introduces the Ising model and the techniques used toavoid the sign problem inherent to the imaginary field. The second part, which contains thenew contributions to the scientific world, applies the aforementioned algorithms to the two- andthree-dimensional cases with success. The complete phase diagram of the model is sketched withthe help of a mean-field calculation and, despite the simplicity of the model, its θ dependenceproved to be more complicated than what one expects for QCD, featuring a phase transition forθ < π for some values of the coupling. The chapter ends with the commitment to apply thesemethods to QCD, in order to find out what would be the behaviour of nature if θ departedfrom zero, and to try to find out what it behind the strong CP problem.

The possibility to overcome the sign problem in the numerical simulations of QCD witha θ term inspired us to deal with another long-standing problem of QCD suffering from thesame handicap: finite density QCD. This difficulties take us to the second part of this work,which deals with the polimerization of fermionic actions. The polimerization as a technique wasborn around thirty years ago, with the purpose of performing simulations of fermions on thelattice. Nowadays, fermion simulations are carried out by means of the effective action, obtainedafter the integration in the full action of the Grassman variables2. This integration yieldsthe fermionic determinant, characterized by its high computational cost, for it usually reacheslarge dimensions3. The polymeric alternative transforms the quite expensive computationallydeterminant of the effective theory into a simple statistical mechanics system of monomers,dimers and baryonic loops, easy to simulate. Although this formulation naively displays asevere sign problem, it can be solved in some cases by clustering configurations, being the mostnotorious one the MDP model of Karsch and Mutter for QCD in the strong coupling limit.

That is why the sixth chapter of this work is completely devoted to the MDP model ofKarsch and Mutter. When this model appeared, it was claimed that it allowed the performanceof numerical simulations of finite density QCD in the strong coupling limit, for the sign problemwithin this model was mild in the worst case. Some doubts were raised when the analysis ofKarsch and Mutter was carefully revised and extended, and in analyzing these objections we

2Direct simulation of the Grassman fields in the computer is not feasible at this moment.3The most impresive numerical simulations involving dynamical fermions invert a ∼ 108× ∼ 108 fermionic

matrix.

iv

found a serious problem with the original simulation algorithm, which was not ergodic for smallmasses. In particular, the chiral limit m → 0 was not simulable at all. Our contribution tothe field is the improvement of the simulation algorithm to enhance its ergodicity, matching thegood properties of the new so-called worm algoriths, to whom it is compared. The results aredisappointing: the sign problem is recovered once the ergodicity is restored, so it was the lack ofergodicity the responsible of the reduction of the sign problem’s severity. This result oppossesthat of Philippe de Forcrand in his last papers, were he claims to solve the sign problem inthe same model by using a worm algorithm. Theoretical arguments explaining why the signproblem for high values of the chemical potential can not be solved within the p.d.f. frameworkare then given.

On the other hand, the MDP model should suffer from a severe sign problem even atzero chemical potential, but from the experience of Karsch and Mutter, it seems that a cleverclustering might be able to remove the sign problem completely, or at least to reduce its severity.This fact takes us to the seventh chapter, which successfully tries to apply the polimerizationtechnique to a pure, abelian Yang-Mills action4, in an attempt to take the MDP model beyondthe strong coupling limit. The polymeric abelian gauge model is simulated and it is foundto perform at least as well as the standard heat-bath procedure, besting it in some areas. Areduction of the critical slowing down for some systems featuring a second order phase transitionis also observed. Nevertheless, the complete system featuring fermions plus Yang-Mills fields hasa severe sign problem, even at zero chemical potential, and can not be simulated in the computer.In the last chapter, which can be viewed as an extension of chapter seven, the polymerizationprocedure is generalized and applied to other systems, mixing failure with success, being themost important of the latter the Ising model under an imaginary field h = iπ/2 for any valueof the dimension, which takes us back to the fifth chapter and the theories with a θ term.

Hence, the underlying topic of these four years of research has been the sign problem inherentto complex actions, in particular finite density QCD and QCD with a θ term, and any deviationfrom this point (mainly chapters two, three, four, seven and eight) is a side-work which ended upgiving interesting results. These are long-standing problems to which hundreds of scientist havedevoted half their lives without too much success. My collaborators and me are no exception,and although some significant advances were made within these four years, we are still very farfrom finding a solution to these problems.

4Although we regarded at first this application of the polymerization as an original contribution never donebefore, we discovered afterwards that we were wrong. The polymerization for abelian gauge theories was proposedsome years ago, but it was never simulated in a computer.

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vi

Part I

Symmetries of QCD and θ-vacuum

1

Chapter 1

QCD and θ-vacuum: the strong CP

problem

“Science never solves a problem withoutcreating ten more”

—George Bernard Shaw

1.1 The symmetries of QCD lagrangian

The full1 QCD lagrangian for NF -flavours is

L =1

4F a

µνFµνa +

N∑

m,n

[iqmγµDµqm − qmMmnqn] (1.1)

where the mass matrix is diagonal and real. The quark fields q belong to the(

12 , 0

)⊕

(0, 1

2

)

representation of the Lorentz group, i.e., they are Dirac spinors. This representation is reducibleand give rise to two sets of Weyl spinors by application of the chiral projectors PL and PR,

qL = PLq qL = qPL

qR = PRq qR = qPR,(1.2)

so we could write the lagrangian in terms of left and right handed fields,

L =1

4F a

µνFµνa +

N∑

m,n

[iqmL γµDµqm

L + iqmR γµDµqm

R − qmL Mmnqn

R] . (1.3)

The key point to notice here is the absence of interactions between the left and the right weylspinors, except for the mass term. In a world of massless quarks, the whole lagrangian wouldenjoy an U(NF )L × U(NF )R symmetry,

qmL → Umnqn

L qmL → qm

L (Umn)†

qmR → Vmnqn

R qmR → qm

R (Vmn)†U, V ∈ U (NF ) , (1.4)

including the so called chiral group, represented by the SU(NF )L × SU(NF )R rotations, andallowing independent transformations of the left and right fields. Nonetheless, this is not thecase of the universe.

1The θ-term will be introduced later.

3

Given an arbitrary diagonal, real mass matrix M with unequal eigenvalues, the only surviv-ing symmetry for each quark field qm is the U(1) phase transformation, associated to fermionnumber conservation:

q′ = eiαq,q′ = eiαq.

(1.5)

As far as we know, the quarks up and down are not massless in Nature, but their masses arequite small, and it is usually a good approximation to work in the massless limit2, where thelight quark masses vanish. The strange quark, although much more massive, usually fits wellin the chiral limit as well. This approximation brings up a U(3)L × U(3)R symmetry group inthe theory, widely decomposed as SU(3)L × SU(3)R × U(1)B × U(1)A. This symmetry is notpreserved: its subgroup SU(3)L × SU(3)R × U(1)V is thought to be spontaneously broken toSU(3)V ×U(1)V , where the V subindex (vector) indicates a coherent rotation in both the L andthe R fermions, whereas the A subindex (axial) denotes an opposite rotation. The chiral limitbecomes a good approximation for those quarks whose mass is much smaller than the QCDscale mq << ΛQCD ∼ 300MeV .

In this context, the SU(2)V ∈ SU(3)V subgroup of the symmetry, associated to rotationsmixing the up and down quarks, amounts to the old isospin symmetry3. In general, the wholeSU(3)V is understood as flavour symmetry : being the three quarks massless in the chirallimit, they become indistinguishable from the QCD point of view; indeed the existence of thissymmetry does not require massless but degenerated quarks. This vector symmetry is expectedto generate multiplets of ‘equivalent’ hadrons related to the different representations of thegroup. The most widely-known are the baryonic decuplet and the meson octet. In fact, this iswhat approximatively happens in nature.

Regarding the remaining symmetries, the U(1)V group entails, as explained above, fermionnumber conservation, and the resting U(1)A is known as axial symmetry, and forces the differ-ence of the number of fermions of each chirality NL−NR to be preserved. As both, NL+NR andNL − NR are to be preserved, we should infer that both NL and NR are conserved separately.However, this is not what actually happens, I will come back to this point later.

The facts supporting the spontaneous breaking of the SU(3)L × SU(3)R symmetry for theup, down and strange quarks are related to the particle spectrum of the world: were the fullsymmetry realized, we should observe in Nature larger ‘equivalent’ particle multiplets, includingpositive parity partners of the standard mesons with similar masses. Up to now, none of thesehave been found, and there are no scalar particles with masses anywhere close to those of thepseudoscalar meson octet.

On the other hand, what we do observe is the existence of light mesons, in this case, thethree pions. These can be understood in the framework of the spontaneous breaking of isospinsymmetry SU(2)L × SU(2)R down to a SU(2)V : the symmetry breakdown generates threemassless Goldstone bosons; as the up and down quarks are not massless in Nature, the pionsare assimilated in the theory as approximate Goldstone bosons, which should be strictly masslessin the chiral limit.

The strange quark enlarges this picture with the four Kaons. These feature masses higherthan the pions, for the strange quark is further from the chiral limit than the up and down

2Also called chiral limit , for at m = 0 the chiral group becomes a symmetry of the Lagrangian again, but thisis not the end of the story. . .

3The wisdom that the isospin (interchange of up and down quarks) is a good symmetry of Nature has a longtradition in nuclear physics. Many relationships among cross-sections have been found using this symmetry.

4

··

···

TTTTT

TT

TTT

·····

r

r

r

r

r rr

r

K0 K+

K− K0

π+π− π0

η-

6

I3

Y

J = 0−

Figure 1.1: Mesonic octet of three flavoured QCD. These are aproximate Goldstone bosonsof the spontaneous breaking of the SU(3)L × U(3)R → SU(3)V symmetry, which leaves eightbroken generators.

quark, but the underlying phenomena giving rise to the low mass of the Kaons is the same.What puzzled the physicists during the 70’s was the absence of a ninth Goldstone boson (ora fourth, in the context of only isospin symmetry with up and down quarks), coming fromthe spontaneous breaking of the U(1)A axial symmetry. Manifestations of this breaking wereobserved in the dynamical formation of diquark condensates 〈uu〉 = 〈dd〉 6= 0. The spectrumof mesons revealed no ninth light meson that could be assimilated to the breaking of the U(1)axial symmetry of QCD. The best candidate was the η′, for it has the right quantum numbersto be the ninth meson, but its mass is too high to be considered a vestige of a Goldstone boson.People began to think that, after all, there was no true U(1)A symmetry in QCD, but this factwas not reflected in the lagrangian. The U(1)A problem was born [1].

1.2 Solution to the U(1)A problem

To understand the current solution to the U(1)A problem we must review the theory a bit moredeeply. Let’s begin with a free spinor of mass m

SF =

∫

d4x(ψγµ∂µψ + mψψ

),

and consider both U(1)V and U(1)A internal transformations of the field ψ

U(1)V

{ψ → eiαψψ → ψeiα U(1)A

{

ψ → eiγ5θψ

ψ → ψeiγ5θ . (1.6)

For small values of α and θeiα ≈ 1 + iα

eiγ5θ ≈ 1 + iγ5θ,

and direct application of the Nother theorem (A.3) produces

U(1)V

jµ = ψγµψ∂µjµ = 0

Q = ψ†ψ = NL + NR

(1.7)

5

U(1)A

jµ5 = ψγµγ5ψ

∂µjµ5 = 2imψγ5ψ

Q5 = ψ†γ5ψ = NL − NR.(1.8)

Therefore, the massless free theory is U(1)A invariant. As jµ and jµ5 are both conserved, the

following currents and charges

jµL = 1

2 (jµ − jµ5 )

jµR = 1

2 (jµ + jµ5 )

QL = NL

QR = NR(1.9)

are preserved, and the number of left-handed and right-handed fermions is constant over time.

What happens if we switch on a gauge field? As our transformation deals only with thefermionic degrees of freedom, the naive conclusion would be that the conservation of bothcurrents holds. Nevertheless we know that there are gauge contributions to the anomaly, whichcan be computed in perturbation theory [2]. In fact, the so-called triangle diagrams

Figure 1.2: Example of triangle diagram contributing to the anomaly.

give rise to the equation

∂µjµ5 = 2imψγ5ψ − NF g2

16π2ǫαβµνF c

αβF cµν . (1.10)

The new term

P = −NF g2

16π2ǫαβµνF c

αβ (x)F cµν (x) (1.11)

is the Pontryagin density P, which encodes the topological properties of the Yang-Mills potentialsand fields, i.e., defects, dislocations and instanton solutions of the gauge fields, deeply relatedto the winding number of the configurations. Thence, chiral symmetry is only preserved at theclassical level, for the quantum corrections of the fermion triangle diagrams break it explicitly.This phenomenon is known as the Adler-Bell-Jackiw anomaly, honoring the discoverers [2].

The path integral formulation

Z =

∫[dAa

µ

]dψdψ e

R

d4xL (1.12)

can also give account of the anomaly. Although the massless action is invariant under a chiraltransformation, the integral measure is not, and its Jacobian

J = e−iR

d4xθ(x)NF g2

16π2 ǫαβµνF cαβ(x)F c

µν (1.13)

reproduces exactly the anomalous contribution of the Adler-Bell-Jackiw triangle diagrams.

6

The chiral transformation introduces a new parameter in the theory, the θ-angle associatedto the chiral rotation, and one might wonder which value θ should take. As the term in the actionarising from the anomaly violates CP -symmetry, most scientist assume θ = 0 on experimentalgrounds4, for as far as we know, CP is a good symmetry of QCD. Nonetheless, there areno theoretical arguments favouring the vanishing value of θ. This mystery, being know as thestrong CP problem, still bewilders the theoretical physicists5.

1.3 The role of the topological charge in the anomaly

An amazing property of this construction is the fact that the Pontryagin index can be writtenas a total divergence

P = −∂µK =

− 1

16π2ǫµαβγ∂µTr

[1

2Aa

α (x) ∂βAaγ (x) +

1

3Aa

α (x)Aaβ (x)Aa

γ (x)

]

; (1.14)

thence, the modification of the action induced by P is a pure surface integral

δS = −∫

d4x∂µKµ =

∫

dσµKµ; (1.15)

which becomes zero when the naive boundary condition Aaµ = 0 at spatial infinity is used.

Furthermore, the redefinition of the axial current as

J µ5 = jµ

5 + Kµ (1.16)

implies that the new current is conserved in the chiral limit

∂µJ µ5 = 2imψγ5ψ =

m→00. (1.17)

Nonetheless, Kµ is not gauge invariant, so in principle, the conservation of the current J µ5

might seem an empty statement. We should wonder, what is the meaning of the equation (1.17)in the chiral limit. There are different points of view to address this phenomenon, being maybethe one introduced by ’t Hooft [3] the most insightful: he showed that the correct boundarycondition to be used is that Aa

µ should be a pure gauge field at spatial infinity, which comprisesAa

µ = 0 and its gauge transformations. But Kµ is not gauge invariant, and there are pure gaugeconfigurations, equivalent to Aa

µ = 0, for which∫

dσµKµ 6= 0.Another way to see it, more practical, is the incompatibility of the regularisation schemes

and the realization of the symmetries of the theory. We usually desire to apply a regularisationscheme which preserves gauge invariance, as this symmetry is what defines the QCD interac-tions. Equation (1.17) implies that this regularisation breaks the chiral symmetry explicitly. Onthe other hand, if we use a regularisation conserving chiral symmetry, we must give up gaugeinvariance. In other words, there is no regularisation scheme that can preserve gauge invarianceand the chiral symmetry at the same time.

As exposed previously, the current J µ5 receives two apparently uncorrelated contributions:

The first coming from the fermionic terms in the lagrangian, jµ5 , is related to the chiralities of

the fermions; and the second one, Kµ, is a pure gauge contribution, associated to the presence of

4The current experimental upper-bound of the θ parameter, based on measurements of the neutron electricdipole moment, is quite small θ ∼

< 10−10 (see [4]).5For a review on this topic, see [5].

7

topological structures in the vacuum. Equation (1.17) relates both through the new conservedcharge of the full symmetry. Integrating (1.17),

Q5 = Q5 + QTop. (1.18)

Thence, what is conserved is the quantity NL−NR+QTop, that simply is the celebrated Atiyah-Singer index theorem. It seems that there is an interplay between the fermionic and the puregauge degrees of freedom, and the change of chirality in the fermions of the vacuum carries amodification of the topological structure of the gauge fields. In fact, there exist solutions forthe gauge fields which are capable of changing the vacua, modifying QTop, and hence NL −NR.This solutions are called instantons, from instant·on, because they appear briefly in time. Theinstantons are quite important to define the theory, for the pure vacua of QCD is not the stateof fixed QTop, but a linear combination of them

|θ〉 =

∫

eiθQTop |QTop〉 . (1.19)

The picture is completely analogous to that of the electrons in a periodic cristalline structure,where θ would be the Bloch momentum and |QTop〉 would represent the position of the electron.

Summing up the former points, it is widely assumed that chiral symmetry is spontaneouslybroken in QCD, giving rise to the octet of light mesons, a remnant of the Goldstone bosonsthat should appear if the mass of the up, down and strange quarks was set to zero. Theninth particle (the η′), too massive to come from a Goldstone boson, adquires its mass fromthe anomaly, which, in the end, is not a true symmetry of the theory, thus has no effects andgenerates no Goldstone boson. Nonetheless, the axial transformation still maps solutions ofthe equations of motion into solutions of the equations of motion. The behaviour of the U(1)A

symmetry reminds to spontaneous symmetry breaking, in the sense that the current J µ5 is

preserved, and the ground state of the theory is degenerated. But in this case the η′ cannot beconsidered as the remnant of a Goldstone boson, for the symmetry transformation associatedto it is not gauge invariant; thus, the different vacua are essentially disconnected. The finalstructure of the theory is a discrete set of equivalent vacua, differing in the values of QTop andNL − NR, but conserving the charge QTop + NL − NR. The system can move through thesevacua through the instanton solutions, which would be (in some sense) an analogous to theGoldstone boson, allowing tunneling between the vacua.

This way, the fluctuations of the topological charge produced by the instantons, those of thetransverse susceptibility6, related to NL −NR, and the chiral condensate are physically related,and the equation relating the transverse susceptibility χ5, the topological susceptibility χT andthe chiral condensate 〈ψψ〉,

χ5 = −〈ψψ〉m

+χT

m2(1.20)

acquires its full sense.

1.4 Discrete symmetries

Any Lorentz invariant local quantum field theories with an hermitian Hamiltonian is CPTinvariant [6]7, where CPT is a combination of three discrete symmetries:

6The transverse susceptibility is defined as χ5 = VhD

`

iψγ5ψ´2

E

−˙

iψγ5ψ¸2

i

, that is, the iψγ5ψ correlation

function integrated to all the points of the lattice.7In the work of Schwinger, the theorem is implicitly proved, as an intermediate step to demonstrate the

spin-statistics theorem. For a more recent and systematic proof, see [7].

8

• Charge conjugation C

This symmetry relates a fermion with its corresponding antifermion.

• Parity P

Parity inverts the spatial coordinates x → −x, that is, it inverts the momentum of aparticle while keeping its spin. Were the universe symmetric under Parity, the physic lawswould not change if the whole universe was reflected in a mirror.

• Time reversal T

The effect of a T transformation is to reverse the time t → −t, flipping spin and momen-tum.

Even if a theory preserves CPT , each one of the symmetries might be violated separately; forinstance, the electro-weak theory, due to its chiral nature, violates the T and CP symmetries.

In the early days of the strong interaction, when most of the work was undertaken byphenomenologist and nuclear physicist, each one of the symmetries (C, P and T ) was thoughto be preserved separately. The aparition of the θ term in the action was quite disturbing,for this term explicitly violates CP symmetry, and such a violation in the strong force wasnot observed. On the other hand, and although the experiments seem to confirm that C, Pand T are good symmetries of QCD, there is no theoretical proof of this fact. A symmetrypreserved at the action level (and including the integration measure of the path integral) mightnot be realized in the vacuum, for spontaneous symmetry breaking might occur. Therefore, andalthough there are convincing indications assuring that these three symmetries are conservedin the strong interaction, this is still unclear from a theoretical point of view. I will deal withthis topic later on, for now, let us go back to the anomaly.

1.5 The anomaly on the lattice

Arguably, the most successful regularisation of QCD so far is the lattice. Its ability to takeinto account the non-perturbative phenomena associated to the strong interactions allows thetheoreticians to explore the low-energy dynamics and spectrum of QCD, an impossible deedto those devoted to QCD perturbation theory. Unfortunately, and in spite of the fact that thediscretization of the pure gauge theory works perfectly, the introduction of the fermions is notso seamless, as the Nielsen-Ninomiya no-go theorem [8] asserts. The theorem states that nofermionic action can verify the four following conditions at the same time

(i.) The Fourier transform of the Dirac operator D (p) is an analytic, periodic function of p.

(ii.) The operator D (p) ∝ γµpµ for a |pµ| << 1. This property forces D to be the fermionicoperator for a Dirac fermion.

(iii.) The operator D (p) must be invertible for an pµ except pµ = 0.

(iv.) The anticommutator{γ5, D

}= 0 vanishes.

The first property is equivalent to require locality for the fermionic action. The second oneforces D to be the the fermionic operator associated to Dirac fermions. As we know, each poleof D accounts for a fermion in the a → 0 limit, thus the third requisite ensures that only onefermion is recovered in the continuum limit. And finally the fourth rule is the realization ofchiral symmetry on the lattice.

9

The naive discretisation of fermions, first proposed by Wilson [9], violates the third pointof the theorem, and its continuum limit produces 2D Dirac fermions. This is not difficult toprove, one just need to write the naive action for a single flavour,

SF

[ψ, ψ

]= a4

∑

x

ψ(x)

(4∑

µ=1

γµUµ (x)ψ (x + µ) − U †

µ (x − µ)ψ (x − µ)

2a

− mψ (x)

)

, (1.21)

with a the lattice spacing. Here, the Dirac operator is

D (x, y) aα

bβ

=4∑

µ=1

(γµ)αβ

Uµ (x)ab δx+µ,y − U †µ (x − µ)ab δx−µ,y

2a

+ mδabδαβδx,y. (1.22)

where a and b are colour subindices, and α and β represent Dirac subindices. The fouriertransform of D allows us to compute the poles of the propagator and see the doublers in thisformulation. This is

D (p, q) =1

V

∑

x

e−i(p−q)xa

4∑

µ=1

γµeiqµa − e−iqµa

2a+ m1

= δ (p − q)

m1 +i

a

4∑

µ=1

γµ sin (pµa)

= δ (p − q) D (p) , (1.23)

and the propagator becomes

D−1 (p) =m1 − i

a

∑4µ=1 γµ sin (pµa)

m21 + a−2∑

µ (sin (pµa))2, (1.24)

which has 2D = 16 poles at m = 0 in the four-dimensional case, given by pµ = (k1, k2, k3, k4),where ki = 0, π

2 .Wilson noticed the problems of the naive action shortly after the publication of [9], and

proposed a simple way to fix it: he added a term to the action, the so-called Wilson term,which is a discretization of − ra

2 ∂µ∂µ, where ∂µ is the full covariant derivative and r is usuallyr = 1:

SF

[ψ, ψ

]= a4

∑

x

ψ (x)

[(

m +4r

a

)

ψ (x)

−4∑

µ=1

(1 − γµ)Uµ (x) ψ (x + µ) − (1 + γµ) U †µ (x − µ)ψ (x − µ)

2a

. (1.25)

The Wilson term generates a mass proportional to 1a to 2D − 1 fermions [10]; therefore the

degrees of freedom associated to these fermions become frozen in the continuum limit, and only

10

a single fermion is recovered. But in doing so, the new action violates chiral symmetry explicitly,even at zero mass of the quark, for the Wilson term mixes chiralities, and behaves in the sameway as a mass term under chiral transformations.

One might think that an explicit violation of chiral symmetry should not be quite prob-lematic, as in the end we have to cope with massive quarks. Sadly, the problem is a bit morecomplicated. An explicit violation of chiral symmetry not induced by the quark mass termsproduces a mixing of the chiral violating operators with the mass terms. In the end we have tosubstract these contributions to the mass terms to find out the right value for the masses of theparticles of our theory, i.e., the masses are renormalized additively, and to worsen the problem,the terms to be substracted diverge, giving in the end a finite contribution. This fact rendersthe measurements of some observables quite difficult. On the other hand, a fermionic actioncomplying with chiral symmetry protect the masses from additive renormalisations, so onlymultiplicative renormalisation occurs, and the relationship between the measured quantities onthe lattice and the physical observables is simplified greatly.

So at first, one is tempted to think that chiral symmetry is quite a desirable property. Indeedit is, and a quite successful action for lattice fermions is the Kogut-Susskind action, also knownas staggered fermions. The problem with the exact realization of chiral symmetry in the latticeis the fact that, as the regulator does not violate chiral symmetry, the anomaly is not accountedfor in the continuum limit. In fact, it seems that there is no simple resolution to this matter: theanomaly requires an infinite number of degrees of freedom to happen, and can not be observedon the lattice. The only way to reproduce the anomaly is use a regulator which breaks thechiral symmetry explicitly on the lattice, and recover it in the continuum limit8.

A great advance was done on this topic by the introduction of the Ginsparg-Wilson fermions[11]. These break explicitly the chiral symmetry in the mildest possible way: the anticommu-tator {γ5, D} equals something that is local,

{γ5, D} = aRDγ5D (1.26)

with R a constant depending on the particular version of the regularisation, and a the latticespacing. This locality allow the fermions to behave as if the anticommutator was realized forlong distances, as we will see in chapter 4. Therefore, the Ginsparg-Wilson regularisation forfermions is capable of reproducing the anomaly in the continuum limit, while keeping the goodproperties of the chiral fermions and keeping the theory free of doublers.

8Actually, some authors [12] claim that the staggered fermions can reproduce the correct continuum limit (andtherefore, the anomaly) by using rooting trick, which escapes the Nielsen-Nimoyima no-go theorem by imposingthe non-locality of the action. In some cases, even numerical agreement has been reached [13], although it is stillunclear if all the aspects of the anomaly are correctly reproduced by the rooting approach [14]. The subject isquite controversial and completely out of the scope of this work.

11

12

Chapter 2

The analysis of SpontaneousSymmetry Breaking phenomena

“A child of five would understand this.Send someone to fetch a child of five.”

—Groucho Marx

2.1 Introduction to Spontaneous Symmetry Breaking (SSB)

In some models, the symmetries of the action (or the Hamiltonian, if we are working withstatistical mechanics models) do not always show up in the ground state, but instead, thesystem may choose among many degenerated ground states, related to each other by the originalsymmetry.

For instance, the spins inside a magnet are free to point everywhere. The Hamiltonianof the system exhibits an SO(3) symmetry, that should lead to a vanishing magnetization.But this scenario only happens at high temperatures, where the spins are disordered. At lowtemperatures, long range ordering appears, and spontaneous magnetization is developed. Itseems that there is a preferred direction for the system, but this is in clear contradiction withthe SO(3) symmetry of the Hamiltonian. What happened to the symmetry? The answer is...nothing. The symmetry is still there, and it is manifested in the degeneration of the groundstate. The different possible ground states are related by the original SO(3) symmetry, whichmeans that if we rotate all the spins at the same time, the energy does not change. Thus thetotal magnetization taking into account all the possible outcomes (directions) is zero.

The following thought experiment might help: imagine that a large set of equal magneticsamples is available to us, and that we can isolate each sample in a different chamber, where nomagnetic fields are allowed. Then we heat each sample in a different chamber, until the magne-tization of each one is completely lost, and then, let them cool down and regain magnetization.At the end, we should find a set of samples whose magnetization vectors are non-zero, but pointrandomly, and averaging magnetizations we would find a vanishing vector.

Strictly speaking, spontaneous symmetry breaking phenomena can only happen in systemswith an infinite number of degrees of freedom. The reason is simple: if the number of degreesof freedom is finite, the system can jump from vacuum to vacuum in a finite amount of time,and in an effective way, the symmetry is realized. If, on the contrary, the number of degrees offreedom is infinite, the degenerated vacua become disconnected and the system is trapped inan asymmetric state. This fact is well known in statistical mechanics systems.

13

For quantum mechanical systems with a finite number of degrees of freedom, the argumentsare a bit more complex: although there are systems in which the different ground states do notenjoy the same symmetry level as the Hamiltonian, all these vacua lie in the same Hilbert space ofsquare-integrable funcions L2

1. The creation and annihilation operators of the vacua are relatedamong each other through unitary transformations. Therefore, the symmetry transformations,which commute with the Hamiltonian by definition, amount to a redefinition of the fields andobservables, there is only just one vacuum and the breaking of the symmetry was impossible toobserve at the physical2 level3.

On the contrary, in theories with infinite degrees of freedom, like quantum field theories(QFT), the different symmetric vacua feature a different representation of the algebra of theoperators. Although every vacua relates to the same dynamical system, their representationsare essentially inequivalent, and cannot be connected by using unitary transformations. In otherwords: the system can not evolve from one vacuum to another, and the SSB is realized4.

In fact, the former though experiment was flawed. If we are patient enough, we will seevariations in the magnetization of the samples as time goes by, even if they are kept at very lowtemperatures. Averaging over an infinite time, we will find a vanishing magnetization vector.The reason that makes the system evolve so slowly is the large number of degrees of freedom ina natural magnet, of the order of Avogadro’s number (NA ≈ 1023). The system can move fromone vacuum to another, but it takes indeed a remarkable long time to do so.

Spontaneous symmetry breaking might happen in system with discrete and with continuoussymmetries. In the case of a continuous symmetry, the Goldstone theorem [16, 17] states that:

In a Lorentz and translational invariant local field theory with conserved currents related toa Lie group G, the spontaneous breaking of the symmetry G generates massless and spinlessparticles called Goldstone bosons, which couple to both, currents and fields.

In simpler words, the breaking of a continuous symmetry implies the existence of masslessparticles [18], one for each generator of the symmetry that does not annihilate the vacuum. Thisset of generators build up the symmetry transformations that relate all the degenerate vacua.

The Goldstone bosons represent long wavelength vibration modes of low-energy. In fact,being the Goldstone bosons massless, it defines a transformation of infinite wavelength whichmodifies the ground state of the system at zero energy cost. But we stated earlier that wheneverSSB occurs, the system is trapped in a single ground state, because this transformation is ill-defined in the Fock space of our theory. Nonetheless, the Goldstone bosons describe and definethe low-energy spectrum of the theory, thus they define the dynamics of the broken phase.

2.2 The p.d.f. formalism

The analysis of the properties of the ground state of a theory can become a tricky exercise. Theground state need neither be unique, nor respect all the symmetries announced in the action

1For instance, one can take two completely different theories, like an Hydrogen atom and an harmonic oscilla-tor, to find that the action of the Hydrogen atom Hamiltonian HH on the harmonic oscillator grond state |0〉HO

is perfectly defined.2See the first chapter of [15].3Another way to look at this property is to use the Feynmann path integral formulation to transform the

quantum mechanical system into a one-dimensional statistical system with a local interaction. A theorem forbidsSSB on such statistical system.

4In this case, the action of the Hamiltonian of a theory in the ground state of a different theory (let’s take,for instance, two Klein-Gordon bosons with different masses), is not well defined. The reason is that the Fockspaces of the two theories are disconnected in the thermodynamic limit, and an expansion of the ground state ofthe first theory in terms of particles belonging to the second theory requires an infinite number of terms.

14

(or the Hamiltonian), for SSB may take place. With a degenerated ground state, the directanalysis becomes complicated and normally offers no useful information, and we need to couplethe system to external sources that break explicitly the symmetry, in order to select a vacuum.Then, we compute the pertinent order parameters of the symmetry for a given value of theexternal field in the thermodynamic limit, and finally we take the zero external field limit, andsee how the order parameters behave. This approach is quite sensible in analytical calculations,and it is, in fact, a standard procedure.

The numerical simulations of statistical systems and QFT ’s (particularly, QCD) share thesame problem. If we try to measure the mean magnetization in a simulation of the Ising model,we find after a large number of iterations that 〈m〉 → 0, even if we are in the ordered phasewhere Z2 is spontaneously broken. The reason is the fact that system is enclosed in a finitevolume, with a finite number of degrees of freedom, and tunneling between the two degeneratedvacua occurs, averaging the magnetization to zero. The easy solution in this case is to measure〈m2〉, which is always non-vanishing at finite volume, but takes very small values (of orderO

(1V

)in the disordered phase, whereas in the magnetized phase

⟨m2

⟩≈ 1. This procedure

might seem quite surprising at first, for⟨m2

⟩is invariant under Z2 transformations, thus it does

not look like a good order parameter for spontaneous Z2 breaking. However, we have to keepin mind that m and its higher powers are intensive operators. As the odd powers of m are goodorder parameters of the Z2 symmetry, in the symmetric vacuum case m → 0 when V → ∞.Since intensive operators do not fluctuate in the thermodynamic limit, m2 → 0 as well.

Unfortunately, this recipe is only valid for the Ising model, and similar models featuring aZ2 broken symmetry. There are other situations, far more complex, where it would be desirableto find a systematic way to deal with the order parameters. The first attemps are a shamelesscopy of the analytical procedure, and basically consist on (i.) adding an external source to theaction, which breaks explicitly the symmetry, (ii.) performing many simulations of the samesystem at different values of the volume and the external field, (iii.) finding the infinite volumelimit for the order parameter at each fixed value of the field, and (iv.) finding the zero field limitof the set of values of the order parameter in the V → ∞ limit. This process involves manysimulations, as we have to take two different limits, thermodynamic and zero external source,and the extrapolations required to reach both limits are usually arbitraly, a fact that enlargesthe systematic errors. In addition, for some systems the external source method introduces asevere sign problem in the simulation. Examples of this are the addition of a θ vacuum term inthe QCD action, or the addition of a diquark source in two-coloured QCD. It seems that theanalytical approach is not so clean when we put it on the computer.

There is an alternative, well-known in statistical mechanics, the Probability DistributionFunction (p.d.f.) formalism. The p.d.f. formalism amounts to the calculation of the probabilitydistribution function P (c) of the interesting order parameter c at vanishing external field. Oncethe function P (c) is known, the mean value of any power of the order parameter cn is easilycomputable as

〈cn〉 =

∫ ∞

−∞dc cn P (c)

In contrast with the external source approach, the p.d.f. requires only one long simulation tomeasure P (c) and no extrapolations at all, so the systematic errors are strongly suppressed.Moreover, external sources introducing a sign problem can be treated normally [20].

Fermions are a specially complicated case. As Grassman variables are not directly simulablein the computer5, the common strategy is to integrate them out, yielding an effective action

5Some efforts in this direction have been done by M. Creutz [19]. Although the results indicate –as expected–an exponential growing of the computation time with the number of d.o.f., the increasing computer power

15

proportional to the determinant of the fermionic matrix. The key point here is the fact that,in many cases, the integration of the Grassman variables is equivalent to an averaging over allthe possible degenerated vacua6; thence, all the order parameters which are fermionic bilinearsvanish by force, configuration per configuration, and the P (c) of the bilinear is not measurabledirectly. The external source method allows the system to accommodate in one of the vacua,breaking the degeneration, but as stated before, even if this procedure is quite standard andwidespread, it is not as clean as we would like.

It would be much better to be able to measure the P (c) for these fermionic bilinears. Ithappens that a way to compute this function was devised fifteen years ago in [21]. Althoughquite advantegeous, the p.d.f. is unfortunately not the standard approach to deal with SSB ofsystems with fermionic bilinears as order parameters. That is why I have considered pertinentto sketch the procedure and introduce the notation in the following sections.

The p.d.f. formalism for fermionic bilinears

Let

O (x) = ψ (x)Oψ (x) (2.1)

be the order parameter of an interesting symmetry of our fermionic system, where O is a constantmatrix, so the only dependence on the lattice site is gathered in the Grassman fields. If thesymmetry is spontaneously broken, we expect the ground state to be degenerate. Each differentvacuum can be labeled with an index α, so the expectation value c of the order parameterbecomes vacuum-dependent cα

cα =1

V

∫

d4xOα (x) . (2.2)

An expression for the P (c) follows from (2.2)

P (c) =∑

α

ωαδ (c − cα) =

⟨

δ

(

c − 1

V

∫

d4xO (x)

)⟩

, (2.3)

where ωα is the weight of the vacuum labeled as α, and the brackets 〈·〉 stands for mean value,computed with the integration measure

[dU ] e−SG det (D + m) ,

which defines the partition function of the theory

Z =

∫

[dU ] e−SG det (D + m) , (2.4)

with (D + m) as the fermionic matrix with mass terms, SG as the bosonic action and U aglom-erating every bosonic degree of freedom.

The rightmost expression of equation (2.3) involves the dirac delta of a fermionic bilinear.This kind of object is –in principle- defined only formally, thus equation (2.3) is empty ofmeaning for fermionic order parameters; were O (x) a bosonic operator, no objections to (2.3)could be raised. The solution to this problem is not straightforward, and involves anotherill-defined step: The computation of the Fourier transform P (q) =

∫dc eiqcP (c),

available for simulations is making this approach interesting for some unsolved scenarios featuring sign problem(i.e. chemical potential, or θ–vacuum).

6Parity is a notable exception, for it acts on the gauge fields as well; ergo the determinant cannot contain allthe possible vacua.

16

P (q) =

∫[dU ]

[dψdψ

]e−SG+ψ(D+m)ψ+ iq

V

R

d4xO(x)

∫[dU ]

[dψdψ

]e−SG+ψ(D+m)ψ

. (2.5)

Again, nothing wrong can be found in this equation for bosonic order parameters, but in thecase of a fermionic bilinear, we have to postulate P (q) directly, skipping (2.3), in order to keepthe discussion rigurous.

After integration of the Grassman variables, the expression for the P (q) is simplified

P (q) =

⟨det

(

D + m + iqV O

)

det (D + m)

⟩

, (2.6)

and the different moments of the p.d.f. are easily calculated as the q−derivatives at the originof the generating function,

〈On〉 =dnP (q)

dqn

∣∣∣∣q=0

. (2.7)

This way we can postulate (2.5) or (2.6) directly as the generation function of all the mo-ments of the distribution function and render the formalism rigorous, even for fermionic fields,and we can define a P (c) as the inverse Fourier transform of the postulated P (q) [21, 20].

We have to take into account that, due to the anticommuting nature of the Grassmanvariables, P (q) is a polynomial of degree the lattice volume V , so all the moments of P (c)higher than V vanish, for they involve an integral of a product of Grassman variables with arepeated index.

17

18

Chapter 3

The Aoki phase

“Research is what I’m doing when I don’tknow what I’m doing.”

—Wernher von Braun

Spontaneous chiral symmetry breaking plays a distinguished role in the properties of QCDat low energies. One expects that the theory of two flavours develops three massless pions anda massive η as the mass of the quarks is driven towards zero: at m = 0 a first order phasetransition takes place, for chiral symmetry is recovered, and then spontaneously broken, givingrise to three Goldstone bosons (the massless pions). Nonetheless the Wilson regularization forfermions breaks chiral symmetry explicitly, and behaves in a very different way. First of all,(i.) the critical line marking the ‘chiral transition’ must be non-chiral, i.e., the critical linecan not mark spontaneous chiral symmetry breaking, for chiral symmetry is violated explicitly.But if the action behaves well (i.e., it leads to QCD in the continuum limit), we expect thepions to become massless at the critical line, whereas the η should stay massive. On the otherhand, (ii.) the critical line lies not at m = 0, but the Wilson r term mixes with the mass term,displacing the critical line towards a value mc (β) which depends on the lattice spacing. Asthere is nothing wrong in using mass values below the critical mass mc, there must be a regionbeyond the critical line with maybe new properties. These two features are combined in the socalled Aoki phase.

The Aoki phase is a region, appearing only1 in actions with Wilson fermions, quenched andunquenched [25, 26, 27, 28, 29, 30, 31], which extends beyond the critical line, and breaks Parityand Flavour symmetries (the latter only in case the number of flavours NF > 1) spontaneously.The usual sketch of the Aoki phase for two flavours2 is shown in fig. 3.1, where the QCDcontinuum-like phase (that conserving Parity and Flavour) is labeled as A, whereas the Aokiphase is marked as B. The Aoki phase develops five fingers at weak coupling, associated to tencritical lines –where the masses of the three pions vanish– which shrink swiftly to isolated pointsas g → 0, leading to different continuum limits. These limits differ in the number of fermions,as can be seen computing the poles of the free propagator: equaling to zero its denominator

(cos (pνaν) − 4 − m)2 + sin2 (pνa

ν) = 0, (3.1)

we find that for each border of the Brillouin zone there is a different continuum limit at a differentvalue of the mass. For m = 0 only the value pν = (0, 0, 0, 0) is a pole of the propagator, so there

1So far, no other regularization has exhibited the same properties. Although in [22, 23] an Aoki phase forstaggered fermions was proposed theoretically, it was never confirmed in simulations. On the other hand, othermodels involving Wilson fermions, but different from QCD, display an Aoki-like phase, see [24].

2The most studied case of the Aoki phase is NF = 2 for obvious reasons.

19

A

A

B

A

κ

1/4

3/8

1/8

8−8/3

0

β

m0

0

35<ιψγ τ ψ> = 0

5<ιψγ ψ> = 0

8

−2

Figure 3.1: Aoki (B) and physical (A) region in the (β, κ) plane. The mean values refer to asingle vacuum, among all the possible ones. Adapted from [41] with courtesy of the authors.

is only one fermion (per flavour) in the continuum limit. However, for m = −2 the condition issatisfied by the vector pν =

(π2 , 0, 0, 0

)and its equivalents

(0, π

2 , 0, 0),(0, π

2 , 0, 0)

and(0, 0, 0, π

2

),

giving rise to four fermions. In the case m = −4 the family of vectors pν =(

π2 , π

2 , 0, 0)

leadsto a continuum limit with six fermions, for m = −6 there are again four vectors similar topν =

(π2 , π

2 , π2 , 0

)and finally at m = −8 only one fermion pν =

(π2 , π

2 , π2 , π

2

)appears as a → 0.

It is clear from this picture that the degenerated two-flavoured action features a symmetry4 + m ↔ − (4 + m), that is, κ ↔ −κ.

For the two flavoured case, Flavour breaking propiciates the apparition of massless particlesin the thermodynamic limit, even if there is no spontaneous chiral symmetry breaking: twomassless Goldstone bosons at the critical line (the two charged3 pions) plus a massless mode as-sociated to the continuous second order phase transition A ↔ B4.The fact that these Goldstonebosons are the pions is related to Parity breaking as well. As we know, the operators associatedto the pions in this case are iψγ5τjψ with j = 1, 2, 3, for the pions are P = −1 particles. Onlythe spontaneous breaking of Parity would allow these operators to develop long-range orderingand non-zero disconnected parts. The η still stays massive, this is reflected in the fact that there

3Which pions become massless depends on the vacuum chosen after the SSB leading to the Aoki phase hastaken place. We will use in this introduction the standard Aoki vacuum, that is, the one selected after addingthe external source hiψγ5τ3ψ.

4Since the transition is continuous, and the two charged pions become massless at the critical line, the neutralpion must also be massless in the transition, so as to recover Flavour symmetry in phase A.

20

are no more Goldstone bosons, and thence there is no massless η. Thence the η is thought notto play further role in the Aoki phase, and the expectation value of any moment of the operatoriψγ5ψ is expected to vanish. In fact, and as stated by S. Sharpe and R. Singleton Jr. in [32],there is a conserved discrete symmetry

P ′ = P × iτ1 (3.2)

which consist of a combination of flavour interchange and Parity transformation, multiplied bya phase to keep the operator hermitian. This symmetry, which is not a replacement for Parity,forces any power of the operator iψγ5ψ to take zero expectation value inside the Aoki phase,even if Parity is broken. The operator ψτ3ψ, which could be used to check Flavour symmetrybreaking, vanishes as well because of symmetry P ′.

Thence, the Aoki phase is characterized by the following properties in the thermodynamiclimit

⟨iψγ5τ3ψ

⟩6= 0,

⟨iψγ5ψ

⟩= 0. (3.3)

This unambiguously implies an antiferromagnetic ordering of the two condensates iψuγ5ψu andiψdγ5ψd, verifying the relation

⟨iψuγ5ψu

⟩= −

⟨iψdγ5ψd

⟩.

The standard picture of the Aoki phase does not allow any other kind of ordering of the P = −1condensates.

3.1 Arguments supporting the standard picture of the Aokiphase

χPT analysis

The theory of chiral Lagrangians indeed supports this standard picture, and is fully compatiblewith the existence of the Aoki phase [32, 33]5. Let us see this at work. When close to thecontinuum limit, the lattice theory can be described by an effective continuum Lagrangian, plussome terms depending on the finite lattice spacing a. Not all terms are allowed to supplementthe original continuum Lagrangian, but only those complying with the symmetries of the latticeaction, i.e., chiral symmetry in this case. Up to first order in a, the result is

LEff ∼ LG + ψ(

D +m0

a

)

ψ − mc

aψψ + b1iaψτµνFµνψ + b2aψ (D + m)2 ψ

+ b3amψ (D + m) ψ + b4amLG + b5am2ψψ + O(a2

), (3.4)

where LG is the pure gluonic Lagrangian, m is the physical mass, which will be defined shortly,and τµν = [γµ, γν ]. In this analysis, there is no attempt to control factors of order unity, likerenormalization factors with logarithmic dependence on a. That is why we use the symbol ∼.

Now we proceed to the analysis of every single term of (3.4). The first two sumands on ther.h.s. correspond to the naive continuum limit of the lattice Lagrangian. The next one is the

5Although the exposition contained here is based in the work of S. Sharpe and R. J. Singleton, Jr., the pioneeron the topic was M. Creutz [34] with an analysis based on the linear σ−model, which predicted the existence ofan Aoki phase as well.

21

most important correction, the additive mass renormalization term, and it is usually absorbedin a redefinition of the mass

m =m0 − mc

a, (3.5)

which is the aforementioned physical mass m. Two comments are in order: (i) the divergence1a is absorbed in the definition, so we are renormalizing the mass term, and (ii) mc is close tomc

(g2

), the mass at which the pion masses vanish, but slightly different. That is why the x

notation is used.

The terms proportional to b2 and b3 vanish by virtue of the leading order equations ofmotion, as one can find a redefinition of the quark variables that makes both terms zero. Theterm proportional to b4 only adds a dependency of the gauge coupling on the renormalizedmass, and the term b5 complicates the dependency of the physical mass on the bare quarkmass. These two last contributions are not relevant for the analysis we are carrying out, so theywill be subsequently ignored. As a result, only the Pauli term b1 survives, and our effectiveLagrangian reads

LEff ∼ LG + ψ (D + m)ψ + iab1ψτµνFµνψ + O(a2

). (3.6)

So what we need to do is to modify the standard χPT Lagrangian to accomodate the Pauliterm.

The standard chiral Lagrangian for two flavours is defined as a SU (2)L ×SU (2)R invarianttheory, describing the low energy dynamics of QCD. Its degrees of freedom are the pion fields,gathered in the SU (2) matrix Σ as

Σ =

(π0√

2π+

π− − π0√2

)

. (3.7)

The Lagrangian of the massless case is simply

Lχ =f2

π

4Tr

(

∂µΣ†∂µΣ)

. (3.8)

The field Σ transforms under the SU (2)L × SU (2)R chiral group as

Σ → LΣR†, (3.9)

where L and R are two independent SU (2) matrices.

In the ground state, the field Σ acquires a non-zero expectation value 〈Σ〉 = Σ0 6= 0, signalingchiral symmetry breaking from SU (2)L × SU (2)R → SU (2)V . The lowest excitations of thevacuum are associated to the Goldstone bosons, in this case, the pions

Σ = Σ0ei

P3j=0

πaτafπ , (3.10)

with τa the Pauli matrices. The addition of a mass term breaks explicitly chiral symmetry. Upto second order in m, this rupture is achieved by adding the potential

Vχ = −c1

4Tr

(

Σ + Σ†)

+c2

16

[

Tr(

Σ + Σ†)]2

, (3.11)

where the mass dependence is introduced in the coefficients c1 ∼ mΛ3QCD and c2 ∼ m2Λ2

QCD.All the terms containing derivatives of the field Σ have been dropped in (3.11): as we are

22

interested in the vacuum state only (not in the dynamics) these terms are irrelevant, thus onlypowers of Tr

(Σ + Σ†) appear6.

The inclusion of the Pauli term does not modify the structure of this Lagrangian, but onlythe value of the coefficients. The reason is the fact that the Pauli term and the mass termtransform in the same way under the chiral group, so the introduction of the Pauli term onlyamounts to a shift in the mass m → m+aΛ2

QCD. Consequently, neglecting terms of order unity,

c1 ∼ mΛ3QCD + aΛ5

QCD c2 ∼ m2Λ2QCD + amΛ4

QCD + a2Λ6QCD.

This mass shift of order aΛQCD due to the Pauli term is expected: for values of m of order a, weshould observe a competition between the explicit breaking of chiral symmetry coming from theWilson discretization, and the breaking coming from the mass term. As we see, terms of orderO

(a2

)are kept in the expression for c2. To be consistent, we should compute all the O

(a2

)

corrections to the underlying Lagrangian, but these turn out to be either negligible (suppressedby higher powers of aΛQCD) or proportional to a2Λ6

QCD, thence the expressions for c1 and c2

remain valid.According to the values of the quark masses, three different possibilities may arise

(i.) The mass take physical values m << ΛQCD and the mass terms dominate c1

and c2: then, c1 ∼ mΛ3 and c2 ∼ m2Λ2 as a → 0, the discretization errors becomenegligible, and the contribution of the c2 term is strongly suppressed, as c1

c2∼ m

Λ . Thesymmetry breaking pattern occurs as in the continuum.

(ii.) The mass is of order O (a) and the mass terms compete against the discretiza-

tion terms in c1 and c2: so the contributions of c2 are strongly suppressed, but c1 isaffected by discretization errors, which amount to a shift in the value of the critical mass.It is usually easier to work with the ’shifted mass’ m′ = m − aΛ2

QCD, that vanishes as c1

vanishes. Then, c1 and c2 have new expressions

c1 ∼ m′Λ3QCD c2 ∼ m′2Λ2

QCD + am′Λ4QCD + a2Λ6

QCD.

(iii.) The shifted mass m′ is of order O(a2

)and the discretization terms dominate:

in such a case, am′ ∼ (aΛQCD)3 and

c1 ∼ m′Λ3QCD c2 ∼ a2Λ6

QCD,

so c1 ∼ c2, the two terms in the potential Vχ are of the same order of magnitude, andthis competition can lead to the Aoki phase, as it is explained in the following lines.

The value of the condensate Σ = Σ0 at the ground state minimizes the potential Vχ. As-suming a general expresion for Σ = A + iτ · B with A2 + B2 = 1, the potential becomes

Vχ = −c1A + c2A2 A ∈ [−1, 1] . (3.12)

This expression is invariant under SU (2)V rotations (L = R in (3.9)), although the B compo-nents of the condensate rotate as a vector. Therefore, a non-zero value of these vector com-ponents of the condensate in the ground state Σ0 = A0 + iτ · B0 indicates Flavour symmetrybreaking SU (2)V → U (1), where the U (1) subgroup of allowed rotations is

eiθτB0 .

6Any term invariant under SU (2)V symmetry can be written as a function of Tr`

Σ + Σ†´

.

23

As Σ0 is an SU (2) matrix, it must verify the condition

A2 + B · B = 1.

Thus, Flavour symmetry breaking occurs if and only if |A0| < 1.

The value of A0 is defined by the potential Vχ, which in turn depends on the coefficients c1

and c2. So depending on the value of these coefficients, Flavour symmetry breaking might ormight not take place. Let us analyze the different possibilities.

There are two scenarios, depending on the sign of c2. If c2 > 0, then the potential is aparabola with a minimum at Am = c1

2c2∼ m′

a2Λ3QCD

, which is a function of the mass m′ and the

coupling g through a. This minimum may or may not lie in the (−1, 1) range; if it is outsidethe range, A0 is forced to take the values ±1, lying at the boundaries, and Flavour symmetry ispreserved. If, on the contrary, |Am| < 1, then A0 = Am and B0 acquires a non-zero value at theminimum. Therefore Flavour symmetry is spontaneously broken to U (1) and the properties ofthe vacuum are those of the Aoki phase.

Let us go deeper into the details of this phase. Without any lose of generality, we assumethat the B0 component of the condensate is aligned to the (0, 0, 1) direction7. Then Σ0 =cos θ0 + iτ3 sin θ0, where

cos θ0 =

−1 Am ≤ −1Am −1 ≤ Am ≤ 1+1 1 ≤ Am

. (3.13)

Thence, inside the Aoki phase the vacuum interpolates smoothly between the two possible valuesof the condensate in the continuum limit. In order to compute the pion masses, we perturbatethe condensate Σ around its equilibrium point Σ0 and work out the value of A. This can bedone expanding equation (3.10), and taking the trace. The result is

A = cos θ0 −sin θ0

fππ3 −

cos θ0

2f2π

3∑

j=1

π2j + O

(π3

j

). (3.14)

Then we put A in the potencial Vχ to obtain

Vχ =

{c2f2

π

(1 − A2

m

)π2

3 − c2A2m + O

(

π3j

)

|Am| < 1c2f2

π(|Am| − 1)

∑3j=1 π2

j |Am| ≥ 1. (3.15)

The coefficient of the quadratic term in the pion fields πj is the physical mass mj ,

m1 = m2 = 0, m3 = c2f2

π

(1 − A2

m

)|Am| < 1

mj = c2f2

π(|Am| − 1) |Am| ≥ 1

. (3.16)

The picture is the following: outside the Aoki phase and the critical line, the three pionsshare the same value of the mass. As they approach the critical line, their masses decrease, untilit vanishes exactly at the critical line. If we cross the critical line, the pions π1,2 remain massless,whereas the pion π3 acquires a non-vanishing mass (this is a consequence of the polarizationof the vacuum in the τ3 direction, and could happen in any other direction). This picture isin complete agreement with the standard expectations on the behaviour of the Aoki phase. Inaddition, the χPT analysis makes some new predictions

7Since most numerical simulations done inside the Aoki phase add an external source term hiψγ5τ3ψ, thisassumption also allows us to connect directly the results exposed here with numerical simulations.

24

1 2 3 1 2 3

1 2

3

-2 -1 0 1 2

ε

0

0.2

0.4

0.6

0.8

1

1.2

1.4

m2 f

2 π /

2c 2

Figure 3.2: Dependence of the pion masses for the case c2 < 0, QCD with an Aoki phase.Figure taken from [32] with courtesy of the authors.

(i.) The width of the fingers in which the Aoki phase occurs is ∆m0 ∼ a∆m′ ∼(

aΛ3QCD

)

.

This result comes from the fact that only in the region where there is a competitionbetween c1 and c2 can an Aoki phase exist8. The result is consistent with the fact that,in the continuum limit, it should not be important if we approach the chiral limit frompositive or negative masses, so the Aoki phase should shrink to a point. Moreover, asthe Aoki phase is not observed in perturbation theory, one expects for the width of thephase a power-law dependency on a. This should hold up to logarithmic corrections inthe lattice spacing (which have been ignored thorough the analysis), when we are closeenough to the continuum limit.

(ii.) The mass of the neutral pion π3 is predicted in terms of quantities computable outsidethe Aoki phase. It can be measured and contrasted against simulations.

(iii.) The expectation value of iψuγ5ψu is connected to the spectral density of the hermitianDirac-Wilson operator γ5D in the origin, through the Banks and Casher relation

⟨ψuγ5ψu

⟩= −πρ (0) .

8In principle, there is no need to worry for higher order terms in the expansion, or the existence of a thirdcoefficient c3. One could argue that, being the two first terms of the expansion comparable, a third term mightbecome important, but a third term would have an order c3 ∼ a3Λ7

QCD, thus well suppressed when close enoughto the continuum limit. A region might arise where these three terms cancel, but it must be a very small region ofwidth ∆m0 ∼ a4, much smaller than the region of the Aoki phase, of order ∼ a3. In the same spirit, an a2Λ2

QCD

contribution to c1 would shift the mass m′ again by O`

a2´

, but the analysis remains the same.

25

In other words, the Aoki phase features small eigenvalues and a closing gap. The calcu-lations reproducing these results can be checked in [32].

The second possibility, c2 > 0, leads to a very different scenario. The parabolic potential Vχ

changes sign and the former minimum becomes a maximum. Therefore, the global minimumwithin the range [−1, 1] lie at the borders, and Σ0 = ±1, according to the sign of m′ (or c1).The expansion of (3.10) in this case is

A = ±(

1 − 1

2f2π

) 3∑

j=1

π2j + O

(π3

j

),

where the sign depends on the sign of m′ or c1. Calling ǫ = c12c2

(the Am of the previous case),the potential around the minima develops the behaviour

Vχ =|c2|f2

π

(1 + |ǫ|)3∑

j=1

π2j + c2 − |c1| + O

(π3

j

), (3.17)

and the three pions share the same non-zero mass

m2jf

2π

2 |c2|= 1 + |ǫ| (3.18)

in a world where Flavour symmetry is not spontaneously broken. Note that the pions neverbecome massless in this case, but there is a minimum value for the pysical mass

m2Min,j = 2 |c2| f2

π . (3.19)

Three comments regarding this analysis are in order:

(i.) This analysis is also valid for improved Wilson fermions, for the discretization errorscontribute as O

(a2

)to c2.

(ii.) The sign of c2 is not predicted by this analysis, and must be found out by numericalsimulations. As the above point stresses, the coefficient c2 might vary from improved tounimproved Wilson fermions, so it might happen that these two cases are different.

(iii.) The results of this analysis rely on the assumption that we are close enough to the con-tinuum limit, so that higher terms in the expansions are negligible.

Numerical simulations

Extensive numerical work was done to find out if the phase diagram of QCD with one andtwo flavours of Wilson fermions coincided with Aoki’s proposal. Most of the computationalresults were obtained by Aoki himself and collaborators during the 80’s and the 90’s. Aoki(and collaborators) succeded in proving that iψuγ5ψu 6= 0 for a single Wilson fermion in thequenched approximation with [35, 36] and without external sources [37]. They also pursueddynamical simulations of two flavours of Wilson fermions with an added external source, andshow numerically the existence of a Parity breaking phase [38, 39], even at finite temperature[40], although in this latter case the Aoki region at some point developed a cusp and was pinchedout in the (β, κ) phase diagram. The standard picture was further reinforced by simulationsby other groups. In particular, the group of E.-M. Ilgenfritz and collaborators found for two

26

-2 -1 0 1 2

ε

0

1

2

3

m2 f

2 π /

2|c

2|

Figure 3.3: Dependence of the pion masses for the case c2 > 0, where no Aoki phase exists.Figure taken from [32] with courtesy of the authors.

flavoured QCD at zero [41] and finite temperature [42] very similar results to that obtained byAoki.

All these simulations (except [37], which is quenched and features a single Wilson fermion)have been done using the external source procedure to analyze the spontaneous symmetrybreaking phenomena. It would be desirable to obtain some results without any contaminationcoming from external sources, and here a problem arises: as in the effective action for fermionson the lattice the Grassman variables are integrated out, the outcoming theory averages overall the possible degenerated vacua of the Gibbs state (i.e., without the external source and inthe thermodynamic limit), which in the case of spontaneous symmetry breaking means thatthe expectation values of (3.3) are averaged to zero in a computer simulation, even insidethe Aoki phase. As explained in the p.d.f. introductory chapter, this is a scenario where thep.d.f. formalism shines, for it allows to compute easily the higher moments of the probabilitydistribution function, which should not be zero if the symmetries are spontaneously broken.

3.2 The p.d.f. applied to the Aoki phase

The Gibbs state, or the ǫ−regime

The introduction of the p.d.f. point of view for the Aoki phase may cast light on the propertiesof the Aoki phase. Our starting point is the computation of the generating function of theselected observable. Here we call Pj(q), P0(q) the generating functions of iψγ5τjψ and iψγ5ψ,which are

27

P1(q) = P2(q) = P3(q) =

⟨∏

j

(

1 − q2

V 2µ2j

)⟩

,

P0(q) =

⟨∏

j

(

1 +q

V µj

)2⟩

, (3.20)

where V is the number of degrees of freedom (including colour and Dirac but not flavour d.o.f.)and µj are the real eigenvalues of the Hermitian Dirac-Wilson operator D(κ) = γ5D(κ), andthe mean values 〈·〉 are computed in the Gibbs state9. Notice that P3(q) has not a definite sign,whereas P0(q) is always positive definite. As the generating functions are computed at zerovalue of the external source, we are at the so-called ǫ−regime (see [43]). This means that thezero-momentum modes of the charged pions are unsuppressed. On the other hand, the neutralpion lies in the p−regime, as mπ3L → ∞, and its fluctuations are completely eliminated.

The q−derivatives of P (q) give us the moments of the distribution P (c). As explainedbefore, the first moment of both distributions vanishes, independently of the realization of thesymmetries. The first non-vanishing moment, in the case of spontaneous symmetry breaking,is the second one. Thus, for the particular case10

iψγ5ψ =1

V

∑

x

iψ(x)γ5ψ(x),

iψγ5τ3ψ =1

V

∑

x

iψ(x)γ5τ3ψ(x), (3.21)

the second moments are

⟨(iψγ5ψ

)2⟩

= 2

⟨

1

V 2

∑

j

1

µ2j

⟩

− 4

⟨

1

V

∑

j

1

µj

2⟩

,

⟨(iψγ5τ3ψ

)2⟩

= 2

⟨

1

V 2

∑

j

1

µ2j

⟩

. (3.22)

In the A region of fig. 3.1 (physical) Flavour symmetry is realized. Thence, the p.d.f. of iψγ5τ3ψ

becomes δ(iψγ5ψ

)and 〈

(iψγ5ψ

)2〉 = 0. For the other order parameter, the equation of thesecond moment

⟨(iψγ5ψ

)2⟩

= −4

⟨

1

V

∑

j

1

µj

2⟩

should vanish in the thermodynamic limit since Parity is also preserved in this region, even atnon-zero lattice spacing. Furthermore, a negative value of 〈c2

0〉 would be quite remarkable, since(iψγ5ψ

)2is the square of an Hermitian operator. This may render unreliable the simulations

far from the continuum limit; we will come back to this point later.

In the Aoki phase (region B) [26] there are vacuum states where the condensate iψγ5τ3ψtakes a non-vanishing vacuum expectation value. This implies that its p.d.f., P

(iψγ5τ3ψ

), is

not a Dirac delta δ(iψγ5τ3ψ

), and therefore 〈

(iψγ5τ3ψ

)2〉 (3.22) does not vanish, due to the

9That is, at zero value of the external source and taking into account all the degenerated vacua.10Here we are abusing a bit of the language. We will keep the notation of (3.21) thorough all this work.

28

apparition of small eigenvalues (near zero modes) of order O(

1V

), which compete against the

volume factors. Indeed this aglomeration of small eigenvalues was recognized long time ago asa signal of spontaneous symmetry breaking in the Banks and Casher formula [44], which relatesthe spectral density of the Hermitian Dirac-Wilson operator at the origin with the vacuumexpectation value of iψγ5τ3ψ [32].

Thence the original vacuum splits, giving rise to the Aoki set of vacua, related among themby P×SU (2) /U (1) transformations. As the other interesting bilinear, iψγ5ψ, is invariant underSU (2) /U (1) transformation and changes sign under Parity, if its expectation value 〈iψγ5ψ〉vanishes in one of the Aoki vacua, it must vanish in any other Aoki vacua. Therefore, andassuming that these are all the degenerate vacua that exist in the Aoki phase, we conclude thatP

(iψγ5ψ

)= δ

(iψγ5ψ

)and 〈

(iψγ5ψ

)2〉 = 0, which implies the following non-trivial relation

⟨

1

V 2

∑

j

1

µ2j

⟩

= 2

⟨

1

V

∑

j

1

µj

2⟩

6= 0. (3.23)

Since the l.h.s. of equation (3.23) does not vanish in the thermodynamic limit inside the Aokiphase due to the presence of small eigenvalues, the r.h.s. must be non-zero as well.

A comment is in order here: the p.d.f. does not allow us to infer the existence of an Aokiphase. It only predicts the expression of the different moments of the distribution function interms of the eigenvalues; in fact, equations (3.22) are valid inside and outside the Aoki phase. Inorder to find out the properties of these expectation values, additional input is needed. Most ofthe time some general properties of the spectrum are enough. In this case, the existence of theAoki phase is assumed, thence (3.22) must be non-zero, so there must exist small eigenvalues∼ O

(1V

), a property which has already been observed in numerical simulations.

Outside the Aoki phase, there is a gap in the spectrum around the origin

|µj | > ǫ ǫ ∈ R+.

Thus the 1V factors kill as V → ∞ the contributions of the inverse of the eigenvalues, all the

expectation values are zero, and Flavour symmetry and Parity are preserved. But as notedbefore, the fact that there is a gap in the spectrum comes not from the p.d.f., but it is deducedfrom the assumption that Flavour and Parity are realized in the vacuum.

Equation (3.23) is not the only non-trivial relation forced by the vanishing of the bilineariψγ5ψ inside the Aoki vacua. According to the well stablished standard wisdom of the Aokiphase, one expects all the even moments of this distribution to vanish

⟨(iψγ5ψ

)2n⟩

= 0, (3.24)

which implies a set of infinite independent relations among the eigenvalues of the HermitianDirac-Wilson operator. The consequence of this set of equations will be reviewed later.

QCD with a Twisted Mass Term, or the p−regime

Non-symmetric Spectral Density of Eigenvalues

Seldom are the numerical simulations inside the Aoki phase performed in the absence of anexternal source. Although this procedure –the addition of an external source– has its draw-backs when ported to numerical simulations, it is a valid theoretical tool to study analyticallyspontaneous symmetry breaking. That is why it might seem interesting to predict, from thep.d.f. point of view, what happens to the Aoki phase when we add an external source like

29

∑

x

imtψ(x)γ5τ3ψ(x), (3.25)

that explicitly breaks Flavour and Parity. For this source, the Flavour symmetry is broken fromSU(2) to U(1).

Let us compute again the p.d.f.’s P0(q) and P3(q) of iψγ5ψ and iψγ5τ3ψ under the effectsof the external source (3.25). Simple algebra gives us the following expressions

P0(q) =

⟨∏

j

(q2

V 2 + 2qV µj

m2t + µ2

j

+ 1

)⟩

,

P3(q) =

⟨∏

j

(q2

V 2 + 2qV imt

m2t + µ2

j

− 1

)⟩

, (3.26)

where again µj are the real eigenvalues of the Hermitian Dirac-Wilson operator and the meanvalues are computed now with the integration measure of the Wilson lattice QCD two-flavouredaction, modified with the symmetry breaking source term (3.25).

The condensates are calculated taking the q−derivatives at the origin of P0(q) and P3(q)

〈iψγ5ψ〉 =2i

V

⟨∑

j

µj

m2t + µ2

j

⟩

,

〈(iψγ5ψ

)2〉 =4

V 2

⟨∑

j

µ2j

(m2t + µ2

j )2

⟩

−

− 2

V 2

⟨∑

j

1

m2t + µ2

j

⟩

− 4

⟨

1

V

∑

j

µj

m2t + µ2

j

2⟩

, (3.27)

and

〈iψγ5τ3ψ〉 =2

Vmt

⟨∑

j

1

m2t + µ2

j

⟩

. (3.28)

Equation (3.28) is well known: taking first the infinite volume limit and then the m → 0 limit,one can reproduce the Banks and Casher result

〈iψγ5τ3ψ〉 = −2πρ (0) , (3.29)

which relates a non-vanishing spectral mean density of the Hermitian Wilson operator at theorigin with the spontaneous breaking of Parity and Flavour symmetries, indicating the presenceof small eigenvalues in the spectrum.

The first equation in (3.27) is actually unpleasant since it predicts an imaginary number forthe vacuum expectation value of a Hermitian operator. However, the issue is solved by using theremnant P ′ symmetry (3.2), which is still a symmetry of the action. The P ′ enforces 〈iψγ5ψ〉 =

0, nonetheless(iψγ5ψ

)2could acquire a nonzero expectation value if P ′ is spontaneously broken.

Concerning the second equation in (3.27), the mt term regularizes the denominators of the

first and second contributions to 〈(iψγ5ψ

)2〉, forcing them to vanish in the thermodynamic limitfor mt 6= 0. The third contribution however, which is negative, vanishes only if the spectraldensity of eigenvalues of the Hermitian Wilson operator ρU (µ) for any background gauge field U

30

is an even function of µ. This is actually not true at finite values of V , and some authors [32, 45]suggest that the symmetry of the eigenvalues will be recovered not in the thermodynamic limit,but only in the zero lattice spacing or continuum limit. If we take this last statement as true,we should conclude:

(i.) The Aoki phase, which seems not to be connected with the critical continuum limit point(g2 = 0, κ = 1/8) [37] is unphysical since the 〈c2

0〉 would be negative in this phase, beingc20 the square of an Hermitian operator.

(ii.) In the standard QCD phase, where Parity and Flavour symmetries are realized in thevacuum, we should have however negative values for the vacuum expectation value of thesquare of the Hermitian operator iψγ5ψ, except very close to the continuum limit. Sincethis operator is related to the η-meson, one can expect in such a case important finitelattice spacing effects in the numerical determinations of the η-meson mass.

Symmetric Spectral Density of Eigenvalues

Assuming that the spectral density of eigenvalues of the Hermitian Wilson operator ρU (µ) forany background gauge field U is an even function of µ, the previous picture changes dramatically.In such a case equation (3.27) forces the vanishing of 〈

(iψγ5ψ

)2〉 for any value of mt

〈(iψγ5ψ

)2〉 = 0. (3.30)

Therefore the p.d.f. of c0 is δ(iψγ5ψ

)and

〈iψγ5ψ〉 = 0, (3.31)

for any mt, and also in the mt → 0 limit. Thence iψγ5ψ = 0 in the Aoki vacuum selected bythe external source (3.25), as stated in [26]; but since iψγ5ψ is Flavour invariant, and changesign under Parity, iψγ5ψ must vanish, not only in the vacuum selected by the external source(3.25), but also in all the Aoki vacua connected to this one by Parity-Flavour transformations.The standard wisdom on the Aoki phase does not contemplate the existence of any more vacuawith different properties, but there is no proof of this point.

In fact, if all the vacua are the one selected by the twisted mass term plus those obtainedfrom it by Parity-Flavour transformations, the spectral density of the Hermitian Wilson operatormust always be an even function of µ, since the eigenvalues of this operator change sign underParity and are invariant under Flavour transformations. Then the spectrum density ρU (µ)must also be symmetric at mt = 0, in the Gibbs state. Now let us come back to expression(3.22), which gives us the vacuum expectation values of the square of iψγ5ψ and iψγ5τ3ψ asa function of the spectrum of the Hermitian Wilson operator, but averaged over all the Gibbsstate (without the external symmetry breaking source (3.25) and averaging over all the vacua).Subtracting the two equations in (3.22),

〈(iψγ5τ3ψ

)2〉 − 〈(iψγ5ψ

)2〉 = 4

⟨

1

V

∑

j

1

µj

2⟩

. (3.32)

This equation would naively vanish, if the spectral density of eigenvalues of the HermitianWilson operator were an even function of µ. Therefore one would reach the following conclusionfor the Gibbs state

〈(iψγ5τ3ψ

)2〉 = 〈(iψγ5ψ

)2〉. (3.33)

31

Nevertheless, S. Sharpe put into evidence in a private communication (developed deeply in [43])an aspect that in [51] we somewhat overlooked: a sub-leading contribution to the spectral densitymay affect (3.33) in the Gibbs state (ǫ-regime, in χPT terminology), in such a way that, not only

〈(iψγ5ψ

)2〉, but every even moment of iψγ5ψ would vanish, restoring the standard Aoki picture.This is equivalent to the imposition of the infinite set of equations (3.24), announced at thebeginning of this chapter. These could be understood as sum rules for the eigenvalues, similarto those found by Leutwyler and Smilga in the continuum [46], which relate the topologicalcharge of the configurations with the eigenvalues of the Dirac operator.

Therefore, and assuming a symmetric spectral density in the thermodynamic limit, there aretwo possibilities for the Aoki phase:

(i.) The standard picture of the Aoki phase is right and complete, the Aoki vacua are char-acterized by

⟨iψγ5τ3ψ

⟩6= 0,

⟨iψγ5ψ

⟩= 0,

and equations (3.24) are verified, thanks to subleading contributions to the spectral den-sity, which conspire to enforce the vanishing of all the even moments of the p.d.f. ofiψγ5ψ. Thence there exists an infinite number of sum rules among the eigenvalues of theHermitian Dirac Wilson operator.

(ii.) The standard picture of the Aoki phase is incomplete, for there exists a set of vacuaverifying

⟨iψγ5ψ

⟩6= 0, completely disconnected from the standard Aoki vacua. As χPT

predicts unambiguously only the standard Aoki vacua, the implications of this secondscenario are quite strong: Chiral Perturbation Theory may be incomplete.

Since the last scenario implies a confrontation with χPT, it is extremely important to findout which one of the two possibilities is realized.

The New Vacua

To understand the physical properties of these new vacuum states we will assume, inspired bythe numerical results reported in the next section, that the spectral density of eigenvalues ρU (µ)is an even function of µ. Then equation (3.33) holds (taking into account the aforementioneddiscussion raised by S. Sharpe), and hence the p.d.f. of the flavour singlet iψγ5ψ order parametercan not be δ

(iψγ5ψ

)inside the Aoki phase, therefore new vacuum states characterized by a

non-vanishing vacuum expectation value of iψγ5ψ should appear. These new vacua can notbe connected, by mean of Parity-Flavour transformations, to the Aoki vacua, as previouslydiscussed.

In order to better characterize these new vacua, we have added to the lattice QCD actionthe source term

imtψγ5τ3ψ + iθψγ5ψ, (3.34)

which breaks more symmetries than (3.25), but still preserves the U(1) subgroup of the SU(2)Flavour. By computing again the first moment of the p.d.f. of iψγ5ψ and iψγ5τ3ψ and takinginto account that the mean value of the first of these operators is an odd function of θ, whereasthe second one is an even function of θ, we get

32

〈iψγ5ψ〉 = −2θ

V

⟨∑

j

−µ2j + m2

t − θ2

(µ2j + m2

t − θ2)2 + 4θ2µ2j

⟩

,

〈iψγ5τ3ψ〉 =2mt

V

⟨∑

j

µ2j + m2

t − θ2

(µ2j + m2

t − θ2)2 + 4θ2µ2j

⟩

, (3.35)

where µj are again the eigenvalues of the Hermitian Wilson operator and the mean values arecomputed using the full integration measure of lattice QCD with the extra external sources(3.34). This integration measure is not positive definite due to the presence of the iψγ5ψ termin the action, but this should not be a problem for the p.d.f. formalism. In fact, the p.d.f.has been previously applied with success to other systems where a sign problem prevented thedirect simulation within an external source [20].

Let us assume that θ = rmt, so after taking the thermodynamic limit the expressions forthe two order parameters

〈iψγ5ψ〉 =

∫2rmtµ

2 − 2rm3t (1 − r2)

(m2

t (1 − r2) + µ2)2

+ 4r2m2t µ

2ρ(µ)dµ,

〈iψγ5τ3ψ〉 =

∫2m3

t (1 − r2) + 2mtµ2

(m2

t (1 − r2) + µ2)2

+ 4r2m2t µ

2ρ(µ)dµ, (3.36)

depend only on mt and not on θ. Here ρ(µ) is the mean spectral density of the HermitianWilson operator averaged with the full integration measure.

Now the mt → 0 limit is equivalent to approaching the vanishing external source (3.34)point in the θ, mt plane on straight line crossing the origin and with slope r. In this limit, theremnant expressions are

〈iψγ5ψ〉 = 2ρ(0)

∫ +∞

−∞

rt2 − r(1 − r2)

(1 − r2 + t2)2 + 4r2t2dt,

〈iψγ5τ3ψ〉 = 2ρ(0)

∫ +∞

−∞

1 − r2 + t2

(1 − r2 + t2)2 + 4r2t2dt. (3.37)

In the particular case of r = 0 (θ = 0) the Banks and Casher formula is recovered

〈iψγ5τ3ψ〉 = −2πρ(0). (3.38)

and the standard Aoki picture is realized, with iψγ5ψ = 0 but for any other finite value of r theflavour singlet acquires a non-zero expectation value proportional to ρ (0). Therefore, if ρ(0)does not vanish, many vacua appear, characterized by a non-vanishing value of the two orderparameters iψγ5ψ and iψγ5τ3ψ. The special case r → ∞ (equivalent to setting mt = 0) can becomputed as well in many ways, and the final result

⟨iψγ5ψ

⟩= 0

⟨iψγ5τ3ψ

⟩= 0

indicates symmetry restoration and confirms a point previously exposed in this chapter: if there

are new Aoki-like phases, characterized by⟨(

iψγ5ψ)2

⟩

= 0, these can only exist in presence

of the standard Aoki phase. As the external source (3.34) does not lead to the Aoki phase ifmt = 0, the symmetries are restored as θ → 0.

33

It must be remarked that the value of ρ(0) could depend on the slope r of the straight linealong which we approach the origin in the θ, mt plane, and therefore, even if results of numericalsimulations suggest that ρ(0) 6= 0 when we approach the origin along the line of vanishing slope,this does not guarantee that the same holds for other slopes. However the discussion in the firsthalf of this section tell us that if ρ(0) 6= 0 at r = 0 (i.e., standard Aoki vacua), ρ(0) should benon-vanishing for other values of r (other vacua should exist).

Quenched Numerical Simulations

The behaviour of the spectral density of the Hermitian Dirac Wilson operator determines whichone of the different scenarios proposed occurs during simulations. It can become symmetricin the thermodynamic limit, or it may be necessary to reach the continuum limit to recoverthe symmetry. In the latter case, the negative expectation value for the operator

(iψγ5ψ

)2

should hinder the measurements of the η−meson mass in the physical region, whereas in thefirst scenario, we find new properties associated to the Aoki phase. These properties comprise,either the existence of phases with non-zero expectation value of

(iψγ5ψ

)2, or the existence of

an infinite set of sum-rules for the eigenvalues of the Hermitian Dirac Wilson operator. Thepossibilities are quite rich, and as far as the author knows, have not been proposed before.

In order to find out the behaviour of the spectral density at the thermodynamic limit (andthus rule out one of the three scenarios), quenched simulations of lattice QCD with Wilsonfermions were performed. After generating ensembles of well uncorrelated configurations for44, 64 and 84 lattices, we diagonalized the Hermitian Wilson matrix for each configuration andmeasured the volume dependence of the asymmetries in the eigenvalue distribution, both insideand outside the Aoki phase.

It is important to point out that, because of kinematic reasons, the trace of the firsts oddp−powers of the Hermitian Wilson operator H(κ) = γ5W (κ) vanish until p = 7, this included.This is related to the properties of the gamma matrices: in order to have a non-zero contribution,a term with an even number of annihilating gamma matrices must occurr. Writting γ5W (κ) asγ5 + γ5κM , with M traceless, any power of γ5W (κ) is readily computed

Tr{

[γ5W (κ)]2n+1}

= Tr

[

γ5

(2n+1∑

i=0

(n

i

)

κiM i

)]

n = 0, 1, 2 . . . (3.39)

The γ5 matrix which multiplies the whole expansion makes all the terms zero, unless there isa power of κM containing a γ5 in the diagonal. The first odd term verifying this conditioncomes from the ninth power of γ5W (κ), in the term κ8M8. This means that the asymmetriesin the eigenvalue distribution of the Hermitian Dirac Wilson operator start to manifest with anon-vanishing value of the ninth moment of the distribution. Fortunately these asymmetries,even if small, are clearly visible in the numerical simulations.

Figs. 3.4-3.8 show the quenched mean value

A(β, κ, mt) =

⟨

1

V

∑

j

µj

m2t + µ2

j

2⟩

Q

, (3.40)

multiplied by the volume for the three different analyzed volumes, in order to see the scaling ofthe asymmetries of the spectrum. As previously discussed, A(β, κ, mt) give us a quantitativemeasure of these asymmetries. We have added an extra V factor to make the plots for thethree different volumes distinguishable: since the value of A(β, κ, mt) is found to decrease asthe volume increases, the plots of the larger volumes are negligible with respect to the plot of

34

the smaller volume 44. Multiplying all the plots by V , they become of the same magnitudeorder.

The mt term in the denominator of (3.40) acts also as a regulator in the quenched approx-imation, where configurations with zero or near-zero modes are not suppressed by the fermiondeterminant11. This is very likely the origin of the large fluctuations observed in the numericalmeasurements of (3.40) near mt = 0 in the quenched case. That is why our plots are cut belowmt = 0.05; in the physical phase, this cutoff is not really needed, but in the Aoki phase it ismore likely to find zero modes which spoil the distribution.

Figs. 3.4 and 3.5 contain our numerical results in 44, 64 and 84 lattices at β = 0.001, κ = 0.17and β = 5.0, κ = 0.15. These first two points are outside the Aoki phase, the first one in thestrong coupling region. The second one intends to be a point where typically QCD simulationsare performed.

0

5e-07

1e-06

1.5e-06

2e-06

2.5e-06

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V·A

(β,κ

,mt)

mt

44

△

△

△

△

△

△

△

△

△△△△△△△△△△△△△△△△△△△△△△△

△△△△△△△△△

△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△

△

64

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbbbbbbbbbbbb

bbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

84

e

e

e

e

e

e

e

e

e

e

eeeeeeeeeeeeeeeeeeeee

eeeeeeee

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

e

Figure 3.4: Point outside of the Aoki phase (β = 0.001, κ = 0.17) and in the strong couplingregime. The superposition of plots clearly states that the asymmetry of the eigenvalue dis-tribution decreases as 1

V . Statistics: 240 configurations (44), 2998 conf. (64) and 806 conf.(84)

Figs. 3.6, 3.7 and 3.8 represent our numerical results in 44, 64 and 84 lattices at β =0.001, κ = 0.30, β = 3.0, κ = 0.30 and β = 4.0, κ = 0.24. These points are well inside theAoki phase, and the structure of the distribution is different from the structure observed inthe previous plots of the physical phase. Nevertheless, the qualitative behaviour as the volumeincreases is the same.

Large fluctuations in the plotted quantity are observed near mt = 0, specially inside theAoki phase. However the behavior with the lattice volume may suggests a vanishing value ofA(β, κ, mt) in the infinite volume limit in both regions, inside and outside the Aoki phase. Ifthis is actually the case in the unquenched model, the spectral symmetry would be recovered

11Although we expect small eigenvalues to appear in the Aoki phase, even with dynamical fermions, in thequenched case these can be ridiculously small [47].

35

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V·A

(β,κ

,mt)

mt

44

△

△

△

△

△

△

△

△

△△△△△△△△△△△△△△△△

△△△△△△△△△

△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△

△

64

b

b

b

b

b

b

b

b

bbbbbbbbbbbbbbbb

bbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

84e

e

e

e

e

e

e

e

e

eeeeeeeeeeeeeee

eeeeeee

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

e

Figure 3.5: Another point outside of the Aoki phase (β = 5.0, κ = 0.15) in a region in whichQCD simulations are commonly performed. The conclusion is the same as in Fig. 3.4. Statistics:400 conf. (44), 900 conf. (64) and 200 conf. (84)

in the thermodynamic limit, the η−meson mass would be measurable reliably at finite latticespacing and the Aoki phase would display new features: either a richer phase diagram or a setof non-trivial sum rules.

A discussion regarding the effect of the regulator mt is in order here. The asymmetry insidethe Aoki phase becomes important for small eigenvalues µ ∼ O

(1V

), but the regulator smooths

out this asymmetry completely. Therefore, and according to our data, the spectral symmetryis recovered for λ ∼> mt. Our regulator was set to mt = 0.05, which is appropiate for V = 44,or might be even for 64, but runs short for 84. On the other hand, the asymmetry for smalleigenvalues is expected, for it is related to the topological charge of the configurations12 [48, 49].A single small eigenvalue crossing the origin might be enough to spoil the symmetry of ρU , hencethe only way to keep the symmetry of the spectral density is that those crossing eigenvaluesbecome zero-modes as V → ∞. As we expect the µ’s to be of order O

(1V

), this should be a

natural consequence. The important point here is the fact that the index theorem is recoveredinside the Aoki phase, in the thermodynamic limit [50].

Unquenched Numerical Simulations

Assuming a symmetric spectral density, we are left with two different possibilities. A briefdescription of each of them follows.

(i.) In the first scenario the conditions

⟨(iψγ5ψ

)2n⟩

6= 0 n ∈ N (3.41)

12This point will be elucidated in detail in the next section.

36

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V·A

(β,κ

,mt)

mt

44

△

△

△

△

△

△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△

△

64

b

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

84

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

e

Figure 3.6: Point inside the Aoki phase (β = 0.001, κ = 0.30) and in the strong coupling regime.Although there is no clear superposition of plots, it is evident that the asymmetry goes to zeroas the volume increases. Statistics: 368 conf. (44), 1579 conf. (64) and 490 conf. (84)

0

0.005

0.01

0.015

0.02

0.025

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V·A

(β,κ

,mt)

mt

44

△

△

△

△

△

△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△

△

64

b

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

84

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

e

Figure 3.7: Point inside the Aoki phase (β = 3.0, κ = 0.30). The asymmetry disappears as thevolume increases. Statistics: 400 conf. (44), 1174 conf. (64) and 107 conf. (84)

37

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V·A

(β,κ

,mt)

mt

44

△

△

△

△

△

△

△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△

△

64

b

b

b

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

84

e

e

e

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

e

Figure 3.8: Point inside the Aoki phase (β = 4.0, κ = 0.24). Same conclusions as in the otherAoki plots. Statistics: 398 conf. (44), 1539 conf. (64) and 247 conf. (84)

are verified as a consequence of the spectral symmetry. To remember how this happens,take the special case n = 2. Then we recover equation (3.22)

⟨(iψγ5ψ

)2⟩

= 2

⟨

1

V 2

∑

j

1

µ2j

⟩

− 4

⟨

1

V

∑

j

1

µj

2⟩

. (3.42)

Inside the Aoki phase, the first member of the r.h.s. of the equation is different from zero,but the second member must be zero (in the V → ∞ limit) due to the spectral symmetry.

Therefore the Aoki phase implies⟨(

iψγ5ψ)2

⟩

6= 0. This point contradicts the standard

picture of the Aoki phase by predicting new phases, unrelated to the original Aoki phase,

and characterized by (3.41). In particular, the prediction⟨(

iψγ5ψ)2

⟩

6= 0 clash to those

of chiral lagrangians, where iψγ5ψ identically vanishes, and thus does any higher powerof it.

(ii.) The second scenario is the one predicted by chiral lagrangians. There,

⟨(iψγ5ψ

)2n⟩

= 0 ∀n ∈ N (3.43)

which in turn implies the existence of an infinite set of sum-rules, a different and inde-pendent one for each n.

These two predictions are incompatible, yet we could not differentiate which one was realizedin our quenched simulations. Therefore, dynamical simulations of the Aoki phase are mandatory.

38

Technical difficulties

Although the simulations of the Aoki phase with dynamical fermions are nothing new in thelattice QCD panorama, these have always been performed under very special conditions: anexternal source is added to the action with a twisted mass term

hiψγ5τ3ψ (3.44)

in order to (i.) regularise the small eigenvalues of the Dirac Wilson operator, which appearonly in the Aoki phase and usually spoil any attempt of simulation without the external source,and (ii.) to analyse the pattern of spontaneous Flavour and Parity breaking. As proved before(3.37) (see [51] as well), the use of an external source like (3.44) selects a standard Aoki vacuum,complying with ((3.43)), whereas an external source

h5iψγ5ψ (3.45)

would introduce a sign problem in the measurement. The only way to investigate the existenceof the new phases characterized by (3.41) is to remove the external source, and extract resultsfrom direct measurements of the Gibbs state (ǫ-regime). The latter point is solved by the p.d.f.formalism, but the former –the removal of the external source– has been an unexplored optionin the Aoki phase dynamical simulations for a number of years. The reason is the appearance ofsmall eigenvalues of the Hermitian Dirac Wilson operator, of order O

(1V

). As far as we know,

the technical problems inside the Aoki phase are similar to those faced when trying to reach thephysical point: The critical slowing down spoils the efficiency of simulations, the Dirac Wilsonoperator becomes increasingly harder to invert, and the performance decreases dramatically. Infact, for some values of the parameters (β, κ) in the coupling-mass phase diagram, the standardalgorithms to invert the Dirac operator just will not work. This fact called for a research oncompetent algorithms to simulate inside the Aoki phase.

Inspiration came in the recent results for new algorithms which reduce the critical slowingdown for small masses in several orders of magnitude. We successfully implemented a SAPpreconditioner in a GCR inverter, as explained in [52]. The new inverter allow us to performsimulations inside the Aoki phase without external sources at a reasonable speed. Unfortunatelythis is not the whole history.

The simulations done outside the Aoki phase feature a spectral gap around the origin, witha symmetric spectrum13. This gap is required to preserve Parity and Flavour symmetry [51].However, the Aoki phase breaks both of them, so this gap was not present in our simulations, andas explained above, the smallest eigenvalues of each configuration took values of order O

(1V

). As

the eigenvalues approach the origin, it may happen that they try to change sign, rendering thespectrum asymmetric, but this movement is forbidden14 by the Hybrid Montecarlo dynamics,i.e., the HMC algorithm is not ergodic for Wilson fermions inside the Aoki phase. Thereforewe are introducing artificial constraints in the dynamics, and the final results are bound tobe modified. That is why we considered another dynamical fermion simulation algorithm, theMicrocanonical Fermionic Average (MFA) [53, 54, 55, 56], which solves the problem of theeigenvalue crossing, but converges poorly as the volume increases.

13By symmetric here we mean that the number of positive and negative eigenvalues are the same.14The movement is not strictly forbidden for higher volumes, as the natural state of the small eigenvalues inside

the Aoki phase is very close to the origin. If the value of the stepsize is high enough, an eigenvalue might crossthe origin. In fact we observed a couple of crossings in our V = 64 HMC simulation, but this transformationis without doubt highly improbable, and constitutes a serious bottleneck in the phase space exploration of thealgorithm.

39

The problem was overcome by using an argument developed in [48, 49] and applied to thecurrent case in [43], where the asymmetry of the spectrum nAsym (the difference between thenumber of positive and negative eigenvalues) is related to the topological charge Q as

nAsym ∝ Q. (3.46)

Then the constraints imposed by the hybrid Montecarlo are equivalent to leaving the topologicalcharge fixed. Since the measurement of the observables should not depend on the value of thetopological charge in the thermodynamic limit, we can select the symmetric sector Q = 0 andmeasure our observables there, where one expects to have smaller finite volume effects15.

Another interesting possibility is (i.) to measure the weight of the different nAsym sectors inthe partition function via the MFA algorithm, (ii.) then perform Hybrid Montecarlo simulationswithin the relevant sectors, and (iii.) do a weighted average of the observables obtained in theHMC using the MFA weights. The results obtained with this method were contrasted to thosecoming from the direct MFA simulations, and with the HMC simulations at fixed nAsym.

A third proposal for simulations could be the addition of a small (3.44) external source, largeenough to regularise the small eigenvalues, but small enough to avoid vacuum selection, wouldallow us to include the eigenvalue crossing phenomenon in our HMC algoritm. Unfortunately,there exists no value of the external field h capable of this two deeds at the same time. If thefield is very small, no vacuum is selected, but the eigenvalues do not cross the origin duringthe simulations; on the contrary, for larger values of the field h, the standard Aoki vacuum isselected. This third possibility was, thus, forsaken.

Numerical results

We performed measurements of three observables of interest

Table 3.1: Expected behaviour of the analysed observables in the different scenarios as V → ∞.

⟨(iψuγ5ψu

)2⟩ ⟨(

iψγ5ψ)2

⟩ ⟨(iψγ5τ3ψ

)2⟩

Outside Aoki ∼ 0 ∼ 0 ∼ 0

AokiStandard Wisdom

6= 0 ∼ 0 6= 0

AokiOur Proposal

6= 0 6= 0 6= 0

Our measurements refer always to the second moment of the p.d.f., which should be zeroin case of symmetry conservation, and non-zero if the symmetry is spontaneously broken [21].The first observable signals Parity breaking, and should be non-zero inside the Aoki phase. Thesecond one allows us to distinguish between our proposal –there is an additional Aoki-like phase

verifying⟨(

iψγ5ψ)2

⟩

6= 0– and the one involving an infinite number of sum rules. Finally, the

third observable is the landmark of the Aoki phase, marking spontaneous Flavour and Paritybreaking.

The first set of simulations were performed using the HMC algorithm, improved with aSAP-preconditioned solver. The symmetric runs nAsym = 0 were performed starting from a

15To clarify this point one could have a look at [46], where the probability distribution of Q is calculatedexplicitly for a finite lattice, and to [57], where the finite volume corrections arising after imposing the constraintQ = cte are computed.

40

cold (ordered, links close to the identity) and from a hot (disordered, close to strong coupling)configuration, obtaining identical results within errors. We could not find an asymmetric statenAsym = 1 in a cold configuration at the values of κ explored, the asymmetric run was startedonly from a hot configuration.

Table 3.2: Results of the Hybrid Montecarlo measurements. NConf indicates the number ofconfigurations.

NConf⟨(

iψuγ5ψu

)2⟩ ⟨(

iψγ5ψ)2

⟩ ⟨(iψγ5τ3ψ

)2⟩

Outside∗

Aoki V = 4420002 2.098(3) × 10−3 4.149(4) × 10−3 4.244(4) × 10−3

nAsym = 0∗∗

V = 4419673 1.90(3) × 10−2 2.69(12) × 10−2 4.90(9) × 10−2

nAsym = 0∗∗

V = 64664 9.2(30) × 10−3 −4.3(30) × 10−1 4.7(30) × 10−1

nAsym = 1∗∗

V = 4410002 6.5(7) × 10−3 −4.5(5) × 10−2 7.1(3) × 10−2

∗Point outside the Aoki phase β = 3.0, κ = 0.22.∗∗Point inside the Aoki phase β = 2.0, κ = 0.25.

We expect all the observables to have non-zero expectation values, even outside the Aokiphase, due to finite volume effects. However, the values inside the Aoki phase are an order ofmagnitude larger than those measured outside the Aoki phase. Outside the Aoki phase, thefollowing approximate rule holds:

2⟨(

iψuγ5ψu

)2⟩

≈⟨(

iψγ5ψ)2

⟩

≈⟨(

iψγ5τ3ψ)2

⟩

,

which is a manifestation of (i.) the presence of an spectral gap and (ii.) the high level ofsymmetry of the spectral density, even at small volumes. This facts can be seen in the p.d.f.expressions for these observables in terms of the eigenvalues µ

⟨(iψuγ5ψu

)2⟩

=1

V 2

⟨N∑

j

(

1

µ2j

)

−

N∑

j

1

µj

2⟩

, (3.47)

⟨(iψγ5ψ

)2⟩

=2

V 2

⟨N∑

j

(

1

µ2j

)

− 2

N∑

j

1

µj

2⟩

, (3.48)

⟨(iψγ5τ3ψ

)2⟩

=2

V 2

⟨N∑

j

(

1

µ2j

)⟩

, (3.49)

where the term

⟨(∑N

j1µj

)2⟩

almost vanishes outside the Aoki phase. Nonetheless, inside the

Aoki phase the gap disappears, the asymmetry near to the origin can become relevant and thisrule would break down.

The numerical results are reported in table 3.2. Some comments are in order: inside the Aokiphase, due to the parity and Flavour breaking, we expect to find

(iψuγ5ψu

)2and

(iψγ5τ3ψ

)2to

be large with respect to the case of standard QCD (outside the Aoki phase), and this is exactely

41

what we get. The crucial point to discriminate between the two scenarios depicted above is thebehaviour of

(iψγ5ψ

)2: if we take the results of nAsym = 0 in the 44 lattice at face value we are

induced to conclude that the non-standard scenario for the Aoki phase is favoured, being the(iψγ5ψ

)2expectation value of the same order of magnitude of the other two observables (and

an order of magnitude larger than outside the Aoki phase). The results of the 64 lattice seemto add no useful informations due to the large statistical errors (we hope to get better qualityresults in the future). On the other hand the result for nAsym = 1 can seem strange at a firstsight (a negative number for the expectation value of the square of an hermitian operator), butwe have to take into account that we are restricting ourselves to a single topological sector andwhat is relevant is the relative weight of the various topological sectors to the final result (thatshould be positive). The evaluation of the weights can not be performed using HMC simulationsdue to ergodicity problems.

We also observed that, in order to achieve in the asymmetric run the same acceptance ratios(∼ 90%) than in the symmetric runs, we had to reduce the simulation step by a factor of ten,for the forces inside the HMC became much larger than expected. We took this as an indicationthat the system was trying to return to the symmetric state, pushing the eigenvalues throughthe origin, thus increasing in an uncontrolled manner the norm of the inverse of the Dirac Wilsonoperator. In the symmetric run, the eigenvalues certainly tried to cross the origin, however thisdid not happen continuously, but only from time to time. Hence, although we can not fully relyin any of these results because of the aforementioned problems of the HMC, we find the resultsof nAsym = 0 much more believable of those of the asymmetric state. Then we introduced theMFA algorithm [53, 54, 55, 56], with the hope of solving the ergodicity problems. As in theMFA the contributions of fermions is added during the measurement, the eigenvalues can crossthe origin at will. So the MFA algorithm allowed us to measure the weights of the differentsectors nAsym = 0, 1 . . ., and then use these weights to correctly average the HMC data.

Table 3.3: Weights of the different sectors according to MFA algorithm.

Volume nAsym = 0 nAsym = 1 nAsym = 2

44 85.9 ± 5.8% 14.1 ± 5.8% 0%

Table 3.4: Results for weighted HMC.

⟨(iψuγ5ψu

)2⟩ ⟨(

iψγ5ψ)2

⟩ ⟨(iψγ5τ3ψ

)2⟩

Weighted HMCV = 44 1.72(9) × 10−2 1.7(6) × 10−2 5.2(3) × 10−2

Conservation of P ′

The numerical results are not conclusive in any way. The expectation value of(iψγ5ψ

)2suffers

from large statistical errors as the volume increases, and we fail to clarify whether the averageis zero or not. We must, either increase our statistics, or find new arguments favouring one ofthe realizations.

The point which favours the standard realization fo the Aoki phase against our proposalis the symmetry P ′. As explained before (3.2), P ′ is a symmetry, composition of Parity P

42

plus a Flavour interchange transformation, which should be conserved inside the Aoki phase,in spite of Parity being broken. This symmetry forces iψγ5ψ to be identically zero, thence⟨(

iψγ5ψ)2

⟩

= 0 and the standard wisdom is realized. An order parameter for P ′ is

X =1

VTr

(γ5D

−1)

=2

V

∑

j

1

µj, (3.50)

which should be 〈X〉 = 0 if the symmetry P ′ is realized. The expectation value⟨X2

⟩is the

second piece of⟨(

iψγ5ψ)2

⟩

. Recalling (3.22)

⟨(iψγ5ψ

)2⟩

=⟨(

iψγ5τ3ψ)2

⟩

−⟨X2

⟩, (3.51)

and we are left with a contradiction: if we are inside the Aoki phase,⟨(

iψγ5τ3ψ)2

⟩

6= 0, and⟨(

iψγ5ψ)2

⟩

would only vanish if⟨X2

⟩takes non-zero values. But certainly

⟨X2

⟩non-zero

implies 〈X〉 6= 0 different from zero as well, breaking explicitly P ′16. In this discussion we areusing the fact that X is an intensive operator which does not fluctuate in the thermodynamiclimit, for it scales with the volume like an intensive operator. Thus we have an strong argumentto support the new scenario for the Aoki phase shown here. Were X not intensive, then thestandard wisdom could be realized.

3.3 Conclusions

Up to now, almost any analysis of the Aoki phase in QCD with two flavours of Wilson fermionsrelied on the addition of a twisted mass term hiψγ5τ3ψ as an external source, in order to breakParity and flavour explicitly, and then take the limit of vanishing external field h → 0, to seewhether the symmetries are restored or not. This approach presents the great handicap ofvacuum selection: the addition of an external source chooses a vacuum where the behaviourof some fermionic observables is predefined. Moreover, the method often requires uncontrolledextrapolations to reach the desired results, and sometimes is not applicable at all because of theintroduction of a severe sign problem in the external source. The p.d.f. formalism avoids thesedangerous steps by computing the Fourier transform of the probability distribution function ofthe interesting fermionic bilinears from the eigenvalues of the Wilson-Dirac operator. As thep.d.f. formalism is unable to predict the form of the spectrum, several possibilities appear.

The first possibility deals with a spectral density ρU (µ, κ) of the Hermitian Dirac-Wilson op-erator, in a fixed background gauge field U , which is not symmetric in µ. This is a true propertyof the spectral density at finite V for the single gauge configurations, even if a symmetric dis-tribution of eigenvalues is recovered by averaging over Parity conjugate configurations. This as-sumption leads to negative values for the square of the flavour singlet iψγ5ψ = iψuγ5ψu+iψdγ5ψd

operator at finite β. Thence, a reliable measurement of the η mass could be only done nearenough to the continuum limit, where the symmetry of the spectral density ρU (µ, κ) should berecovered. Furthermore assuming that the Aoki phase ends at finite β, as some authors suggest[45, 58], the physical interpretation of the Aoki phase in terms of particle excitations would belost. Quenched computer simulations at several small volumes (44, 64 and 84) seem to rule outthis option. This arguments strongly suggest that the second possibility is realized.

In the second scenario, the symmetric spectral density ρU (µ, κ) is assumed to become sym-metric in the infinite volume limit. Under this assumption, the existence of the Aoki phase

16There is no contradiction outside the Aoki phase, for all the observables take zero expectation values in thethermodynamic limit.

43

as established by the common lore is linked to the appearance of other phases, in the sameparameters region, which can be characterized by a non-vanishing vacuum expectation value ofiψγ5ψ, and whose vacuum states are not connected to the Aoki vacua by Parity-flavour sym-metry transformations. This scenario is quite unexpected: Aoki’s approximated calculations[27, 28] of iψγ5ψ reveal that this observable vanishes all along the Aoki phase, and χPT sup-ports unambiguously this picture. Nevertheless, the p.d.f. calculations are exact, hence, (i.)either the approximations used in all the previous calculations of the two-flavoured Aoki phaseare not accurate enough to predict iψγ5ψ 6= 0 and χPT is either incomplete or not the righttool to analyze the Aoki phase, or (ii.) there must be a way to reconcile the results of thep.d.f. with those of previous calculations. Indeed a way exists, but involve the realization of aninfinite set of sum rules.

The χPT practitioners strongly defend their results [32, 33], which predict two possiblerealizations for lattice QCD with two flavours of Wilson fermions at small values of hte latticespacing a. These depends on the sign of a coefficient c2 in a expansion up to second order of thepotential energy of the effective Lagrangian. The first possibility c2 > 0 predicts the standardAoki phase, with spontaneous Parity and Flavour breaking, whereas in the second realizationc2 < 0 the three pions remain massive (in lattice units of mass) all thorough the parametersspace. The sign of c2 depends on the lattice action, thus χPT states that unimproved andimproved actions might behave differently, and that the Aoki phase might not be present forsome versions of the QCD action with Wilson fermions.

A tentative way to harmonise the χPT results with the p.d.f. was devised in [43], where areason why the sum rules must hold is exposed. In simple words, the existence of an unbrokenflavour-rotated Parity symmetry, P ′ = −iτ1 × P , is proposed [32], and this symmmetry forces

iψγ5ψ to vanish. On the other hand,⟨(

iψγ5ψ)2

⟩

is composed of the difference of two pieces,

both of them strictly non-negative. The first piece is always non-zero inside the Aoki phase,whereas the second piece W 2 can be understood as the square of an intensive hermitian operator,which is an order parameter for the P ′ symmetry. The realization of the χPT scenario wouldrequire W 6= 0 to compensate the first piece and let the sum rule be fulfilled. However, W 6= 0implies that the P ′ symmetry is violated, reaching a contradiction.

At this point we considered to carry out unquenched simulations of the Aoki phase withoutthe aid of external sources. The aim of these simulations is twofold: (i.) to elucidate theconservation or violation of P ′, and (ii.) to measure the expectation value of the interestingobservables in the Aoki region. In addition, these simulations are completely new in the QCDpanorama (former simulations inside the Aoki phase for two flavours have always been done witha twisted mass term), and suffer from a critical slowing down, quite similar to that of simulatingsmall masses. In spite of the utilization of a recently developed SAP solver [52] to overcome theslowing down, another severe problem arised, related to the inability of the Hybrid Montecarlo(HMC) algorithm to cross eigenvalues through the origin, due to an appearing divergence inthe fermionic force. Thus, the HMC algorithm is not ergodic inside the Aoki phase, in theabsence of external sources which regularise the unbounded eigenvalues. The MicrocanincalFermionic Average (MFA) represents a valid alternative to HMC, capable of overcoming theergodicity problems, although it displays poor convergence for higher volumes. Unfortunatelythe simulations turned out to be noisy enough to prevent us to extract conclusive results. It

seems that⟨(

iψγ5ψ)2

⟩

certainly takes non-zero values for V = 44, but the scaling of this value

with the volume remains unclear.

44

Chapter 4

Realization of symmetries in QCD

“In all science, error precedes the truth,and it is better it should go first thanlast.”

—Hugh Walpole

4.1 Parity realization from first principles in QCD

Parity P is widely regarded as a symmetry of QCD by the scientific community. There isnothing wrong with this assumption: Indeed Parity is a symmetry of the QCD action in theabsence of a θ-term –whose upper bound, established by experimental results, is extremelysmall–, and experimental data suggest that Parity is not spontaneously broken, at least at theenergies investigated. Nonetheless, a proof of Parity realization in QCD was remarkably absent.In fact, it was revealed in lattice simulations with Wilson fermions that, for some values of theparameters, Parity could become spontaneously broken. Being lattice QCD the most successfulnon-perturbative tool to analyze QCD at low energies, this fact was quite disturbing.

During the mid-eighties, Vafa and Witten published a couple of theorems [59, 60] whichseemed to aliviate the concerns of the theoretical people on this matter. These theorems provedthat neither Parity nor any vector symmetry (the latter includes Flavour if there is some de-generation in the quark mass) were spontaneously broken in a vector-like theory. Sadly, thistheorems were far from perfect, but the effort put into them was not wasted, as they werevery important steps in the development of more refined proofs. In the following sections I willreview them carefully, point out the flaws, and fix them as well as I can.

Review of the Vafa-Witten theorem

The original theorem was unanimously praised by the scientific community, for it provided avery elegant proof of Parity realization, which required only two pages of explanation. Thearticle was sort and beautiful, and the underlying ideas were brilliant; however the proof wasnot complete.

The core of the proof is the fact that, if the vacuum expectation value of any P-nonconservinghermitian operator X vanishes, then the underlying theory preserves Parity. Let’s assume thatX is any Lorentz invariant, hermitian, P-nonconserving operator, composed of Bose fields, andlet L (λ) the original lagrangian of the theory L, plus an external source term L (λ) = L− λX.If λ ∈ R, then the new lagrangian is hermitian. The energy E (λ) of the ground state is easilydeduced to lowest order in λ

45

E (λ) = E (0) + λ 〈X〉

where 〈X〉 is the vacuum expectation value of X at λ = 0. As X is not invariant under P ,the scalar 〈X〉 can take either sign, in particular the one where λ 〈X〉 is negative, lowering theenergy of the ground state. Therefore, if P is violated, there is at least one operator X verifying〈X〉 6= 0, and for a small non-vanishing λ this implies E (λ) < E (0).

The way to rule out this possibility requires of the path integral formulation in the Euclideanspace, but first the operator X must be Wick-rotated. As required before, X is an hermitianBose scalar. Therefore it must be a combination of the gauge fields Aa

µ, the metric tensor gµν

and the antisymmetric tensor ǫµνρσ. The latter assures that X is P -nonconserving, as long asit appears as an odd power in X. The fields Aa

µ and the tensor gµν remain real in both, theEuclidean and the Minkowski space, but the tensor ǫµνρσ picks up a pure imaginary factor i inthe Wick rotation. As X is proportional to an odd power of this tensor, the operator X itselfpick up an i factor as well. The free energy E (λ) then becomes

e−V E(λ) =

∫

dAaµ dψ dψ e−

R

d4x(L+iλX), (4.1)

where the acquired i factor of X has been made explicit. Integrating out the fermions

e−V E(λ) =

∫

dAaµ det (D + m) e−β

R

d4xTrF aµνF µν

a eiλR

d4xX , (4.2)

where (D + m) is the fermionic matrix with a mass term. Since the determinant det (D + m) isdefinite positive in vector-like theories, the positivity of the r.h.s. of (4.2) can only be spoiled bythe term eiλ

R

d4xX . This is a pure phase factor, as the hermiticity of X assures that the integral∫

d4xX is real. In fact, the pure phase factor is the only contribution of the external source λXto the free energy. Since a pure phase factor can only decrease the value of the integral, themaximum of e−V E(λ), and thence the minimum of E (λ), lie at λ = 0. Therefore, 〈X〉 = 0 andParity is preserved. As a side result, Vafa and Witten state that the same argument could beapplied to the QCD lagrangian in the presence of a θ-term, to conclude that the minimum of thefree energy is achieved at θ = 0. For other discrete symmetries (charge conjugation C and timereversal T ), the argument fails. An extension to fermoinic order parameters is straightforward,according to the authors.

Objections to the theorem

1. The proof is not valid for fermionic order parameters

The proof is so simple and elegant, that it is difficult to find arguments against it. Perhaps,the firsts to notice that something was wrong with so much elegance were S. Sharpe andR. Singleton Jr., in its analysis of the Aoki phase –a quite notable counter-example of theP -preserving theorem– published in 1998 [32]. It is quite remarkable that 14 years wereneccessary for the scientific community to acknowledge that the theorem was flawed.

The remark of Sharpe and Singleton is as simple as deadly for the proof. The theorem hasbeen developed for bosonic operators, but Vafa and Witten argue that the extension toFermi fields is straightforward: For instance, let us assume that there are two degeneratedflavours in our action, and we couple to an external field the P -violating operator X =iψγ5τ3ψ (x). After the integration of the Grassman variables, X becomes

X = Tr [iγ5τ3SA (x, x)] ,

46

where SA (x, y) is the fermionic propagator in the background gauge field A. As X isonly a function of A, we can apply the argument safely. This is not strictly right: In thepresence of fermionic zero-modes, the operator X is ill-defined, and in any other case,since the integration over the Grassman fields averages over all the possible degeneratedvacua, X always vanishes, and the eigenvalues of γ5τ3SA (x, x) come in opposite pairs.

One way to overcome the vacua averaging after the integration of Grassman variables isthe addition of a fermionic external source to the original Lagrangian, which breaks Parityexplicitly, and then taking the zero external field limit. Then, the fermionic determinantis modified in the integration measure. In order to evaluate the behaviour of the orderparameter in the ground state, the determinant must be expanded in small powers of theexternal field. However, not all the terms of the expansion are odd under Parity, butsome of the terms become P -even. As these terms are even under Parity transformations,they do not become purely imaginary after a Wick rotation, and do not amount to a purephase. Therefore, the argument is not valid for fermionic order parameters.

The clearest way to see it is the fermionic determinant, as first T. D. Cohen pointed outin [61]. Using the last example, it is easy to compute the new determinant

det (D + m − iλγ5τ3) =∏

n

(µ2

n + h2)≥

∏

n

µ2n = det (D + m) , (4.3)

with µn the n eigenvalue of the fermionic matrix, including mass terms. The addition of theexternal source obviously decreases the value of the energy E (λ) < E (0), in contradictionwith the Vafa-Witten theorem. The failure is evident, as Xiangdong Ji shown in [62], ifwe expand the determinant as

det (D + m − iλγ5τ3) = eTr ln(D+m−iλγ5τ3) =

det (D + m) eTr ln[1−(D+m)−1iλγ5τ3], (4.4)

so the gluonic operator we have to consider is

Tr ln(

1 − (D + m)−1 iλγ5τ3

)

, (4.5)

which has Parity-even pieces, contributing to the lowering of the energy. Indeed theAoki phase is a well documented counter-example of the applicability of this theorem tofermionic bilinears.

2. The free energy is ill-defined

Only a year later, V. Azcoiti and A. Galante pointed out [63] that the theorem was onlyvalid if the free energy was well-defined in the presence of the P -breaking term λX. Infact, it is quite surprinsing that the realization of a symmetry depends on the fact thatthe order parameter X picks up an imaginary i factor under a Wick rotation. As stressedat the beginning of this chapter, although the external source method is an useful toolto analyze the spontaneous breaking of symmetries, the p.d.f. is also a valid formalismto deal with symmetries, so all the information required to know whether a symmetry isspontaneously broken or not is encoded in the ground state, and there is no need to addany external sources.

Let us see the claim of Azcoiti and Galante at work. Here, X is a lorentz invariant,bosonic, hermitian, local operator, verifying the same properties as in the original article

47

of Vafa and Witten. In order to operate with well-defined mathematical objects, a lat-tice regularization which ensures the positivity of the determinant (i.e. Kogut-Susskindfermions) is chosen. In addition, since the operator X is intensive, it is assumed that itdoes not fluctuate when the system rests in a pure vacuum state. In other words: All theconnected correlation functions are supposed to verify the cluster property.

Taking into account these assumptions, the partition function of the generalized La-grangian can be written as a function of the p.d.f. of X as

Z (λ) = Z (0)

∫ ∞

−∞dX P

(

X, V)

eiλV X , (4.6)

with P(

X, V)

the p.d.f. of X at finite volume.

Now let us assume that Parity is spontaneously broken. Then we will show that thevacuum expectation value 〈X〉 is ill-defined in the thermodynamic limit. The vacuumstructure of a theory with spontaneous Parity breaking can be as complex as one canimagine; the only requisite is that any of the degenerated vacua is related to anotherdifferent vacuum by the P operator. Here we will assume that the vacuum structure isthe simplest possible which breaks Parity, i.e., a couple of symmetric Dirac deltas,

limV →∞

P(

X, V)

=1

2δ(

X + a)

+1

2δ(

X − a)

, (4.7)

in the thermodynamic limit, which will translate into a two peak structure with centersat a and (−a) at any finite value of the volume.

As the P symmetry becomes a Z2 symmetry for scalars like X, we can take advantage ofthis and write

Z (λ) = 2Z (0) Re

[∫ ∞

0dX P

(

X, V)

eiλV X

]

, (4.8)

for P (x, V ) = P (−x, V ). Then we take out a factor eiλV a

Z (λ) = 2Z (0) Re

[

eiλV a

∫ ∞

0dX P

(

X, V)

eiλV (X−a)]

, (4.9)

and operate, expanding the exponential in sines and cosines. In the end we arrive at

Z (λ)

2Z (0)= cos (λV a)

∫ ∞

0dX P

(

X, V)

cos[

λV(

X − a)]

−

sin (λV a)

∫ ∞

0dX P

(

X, V)

sin[

λV(

X − a)]

. (4.10)

The zeroes of the partition function are easily obtained from (4.10), equalling the r.h.s.to zero. The resulting equation can be expressed as

cot (λV a) =

∫ ∞0 dX P

(

X, V)

sin[

λV(

X − a)]

∫ ∞0 dX P

(

X, V)

cos[

λV(

X − a)] . (4.11)

48

Up to now the discussion is quite general for any landscape pattern of Parity breaking, in

the sense that this equation is valid for any P(

X, V)

. The only mention to the double-

peaked structure anticipated earlier in (4.7) is the apparition of the center of the peak a,which is not strictly neccessary. Now we need the structure defined in (4.7) to carry onour analysis. Although we expect the same behaviour for other patterns, the aim pursuedhere is to reveal a counterexample for the P -conservation theorem, thus it is enough toshow what happens to the simplest case.

It is easy to see that the denominator of the r.h.s. of (4.11) is constant as V → ∞

limV →∞

d

d (λV )

∫ ∞

0dX P

(

X, V)

cos[

λV(

X − a)]

=

limV →∞

∫ ∞

0dX P

(

X, V) (

X − a)

sin[

λV(

X − a)]

= 0, (4.12)

as P(

X, V)

becomes a Dirac delta and the factor(

X − a)

makes everything vanish. The

same result applies to the numerator of the expression, but the l.h.s. of (4.11) oscillateswildly, giving rise to an infinite number of solutions in the thermodynamic limit. Thusthere are an infinite number of zeroes approaching the origin (i.e., λ = 0) with velocity V ,and the free energy does not converge1. For instance, suppose the double peaked structureis gaussian at finite volume

P(

X, V)

=1

2

(V

π

) 12 (

e−V (X+a)2

+ e−V (X−a)2)

. (4.13)

Then, the partition function becomes

Z (λ) = Z (0) cos (λV a) e−λ2

4V , (4.14)

and the expectation value of the order parameter is

i 〈X〉 =λ

2+ a tan (λV a) , (4.15)

which oscillates wildly for a 6= 0.

This discussion might seem somewhat artificial, but in fact very simple models fittingin Vafa and Witten’s theorem requirements break down in the presence of a P = −1operator. One of the simplest examples is the Ising model with an imaginary magneticfield, which breaks the Z2 symmetry at low temperatures, in clear contradiction with theresult of the theorem. Under these conditions (low temperature, external imaginary field),the free energy is ill-defined in the thermodynamic limit. Although the Ising model is nota vector-like theory, it verifies all the assumptions required for the theorem, even thoseassociated to a vector-like theory (positivity of the measure).

The theorem is only valid if the free energy is well-defined, which implies that the sym-metry is realized in vacuum, so it becomes a tautology.

1The result derived here can be understood as a generalization of the Lee-Yang theorem [64] for any systemfeaturing a Z2 symmetry.

49

3. The argument fails at finite temperature

In 2001 T. D. Cohen cleverly remarked in [61] a number of loopholes in original Vafa andWitten reasoning. The expansion of the free energy

E (λ) = E (0) + λ 〈O〉λ=0

is assumed to hold always, but this point must be treated with care. The limits λ → 0and V → ∞ are not interchangeable, and this linear expression might not be valid in thethermodynamic limit. For instance, the free energy

E (λ) = E0 +√

αV −βSpace + γλ2

with α, β and γ constant and positive, has a minimum at λ = 0, even in the infinitevolume limit, but as V → ∞ it develops a cusp at λ = 0, which signals spontaneoussymmetry breaking (the first derivative becomes discontinuous). As Cohen explains, onecan rule out this possibility on physical grounds: This kind of behaviour is related tolevel-crossing, and level-crossing can never happen at a global-minimum. Nonetheless thepicture changes at finite temperature.

The limitation of the temporal length of our box does not seem to affect to Vafa and Wittenreasoning, and the fermionic determinant remains non-negative. Thence the argumentshould be valid for finite temperature, and Lorentz-invariant P -violating operators havevanishing vacuum expectation values. But the argument does not apply to Lorentz-noninvariant operators. As the heat-bath of finite temperature introduces a rest frame, itviolates Lorentz symmetry, and allows the introduction of the velocity four-vector uµ inthe operators. Thus, observables like

Tr[(

~D × ~E)

· ~E]

= ǫαβγδTr [DαFγσFδρ] uβuσuρ

are not purely imaginary in the Euclidean space, and they do not become a pure phase inthe path integral. The Vafa-Witten theorem is useless here.

4. P -breaking gluonic external sources always increase the free energy

This loophole was exposed by Xiangdong Ji in [62], and may invalidate Vafa and Witten’sParity theorem. The argument is the following: The addition of an external source neednot decrease the energy of a system, regardless of the realization of the symmetry. A veryclever example –which is not completely analogous, and must be understood with care– isthe antiferromagnetic Ising model. In the presence of an external magnetic field, the Z2

symmetry is explicitly broken, nonetheless the energy of the vacuum in the ordered phaseincreases.

Of course, the problem with the Ising example is the election of the right order parameterfor the symmetry, for not every symmetry breaking term selects a vacuum effectively.The point with Parity-odd gluonic operators, according to Xiangdong Ji, is that theycan not select a vacuum either: They contribute as a pure phase factor, whose phaseis proportional to the volume in the path integral. The factor oscillates wildly in thethermodynamic limit, sampling over all the possible degenerated vacua, and erasing anyeffects of the breaking of the symmetry by the external source.

The solution proposed by Xiangdong Ji is, in some sense, analogous to the role played bythe Wick rotation in the path integral formulation. If we allow the external field to be

50

purely imaginary, the phase factor becomes an exponential, capable of selecting a vacuum.Then, we can asses the effects of the symmetry breaking by the external source.

This is only an incomplete sample of the claims of the wrongness of Vafa and Witten’s resulton Parity, but there are much more (see, for instance [65, 66]). On the whole, the Vafa andWitten result [59] can not be considered as a true theorem, and maybe the clearest example isthe existence of a Parity breaking Aoki phase for two flavours of Wilson fermions, in spite ofthe theory complying with all the requirements of the theorem.

The p.d.f. approach to the problem

Although the arguments against Vafa and Witten theorem are quite serious, they do not rule outParity conservation in vector-like theories at all, but only state that the theorem is incorrectand must be improved or reformulated. Most of the claims should be solved if the proof isbased on the p.d.f. formalism, where the degeneration of the vacua is explicitly signaled in theprobability distribution function of each observable. Even the point of fermionic bilinears canbe addressed by using the extension of the p.d.f. developed at the beginning of this chapter.

So let us see now an alternative way of reaching the Vafa-Witten result [67], that makes useof the concept of the p.d.f. of a local operator. I will assume hereafter that a quantum theorycan be consistently defined with a P -breaking local order parameter term.

Let be Y (Aaµ) a local operator constructed with Bose fields. The probability distribution

function of this local operator in the effective gauge theory described by the partition function(2.4) is

P (c) =

⟨

δ

(

c − 1

V

∫

d4x Y (x)

)⟩

(4.16)

where V is the space-time volume and the mean values are computed over all the Yang-Millsconfigurations using the integration measure

[dAa

µ

]eSB det (D + m) ,

One can define the Fourier transform of the p.d.f. as

P (q) =

∫

dc eiqcP (c) (4.17)

which, in our case, is given by the following expression

P (q) =

∫ [dAa

µ

]e−SPG+( iq

V−λ)

R

d4x Y (x)+ln det(D+m)

∫ [dAa

µ

]e−SPG−λ

R

d4x Y (x)+ln det(D+m)(4.18)

or, in a short notation:

P (q) =⟨

eiqV

R

d4x Y (x)⟩

(4.19)

The distribution function P (c) of a local P -breaking operator in absence of a P -breaking termin the action (λ = 0) should be a Dirac delta distribution, centered at the origin, only if thevacuum state is non degenerate, so Parity is preserved. On the contrary, if Parity is broken,the vacuum state is degenerate, and the expected form for P (c) is

P (c) =∑

α

wα δ (c − cα) , (4.20)

51

where cα is the mean value of the local order parameter in the vacuum state α and wα arepositive real numbers which give us the probability of each vacuum state (

∑

α wα = 1).

When the degeneration of the vacuum is due to the spontaneous breaking of a discrete Z2

symmetry like Parity, the system is likely to follow the more standard case of two symmetricvacuum states (±cα) with the same weights wα (or an even number of vacuum states withopposite values of cα in the most general case). The probability distribution function P (c) willbe then the sum of two symmetric Dirac delta’s with equal weights:

P (c) =1

2δ (c − cα) +

1

2δ (c + cα) (4.21)

and its Fourier transform

P (q) = cos (qcα) (4.22)

which can take both positive and negative values.

The relevant fact now, as stated in Vafa-Witten’s paper, is the fact that the local P -breakingorder parameter is a pure imaginary number Y (Aa

µ) = iX(Aaµ))2. In such a case, the P (q),

P (q) =⟨

e−qV

R

d4x X(x)⟩

. (4.23)

which is real and positive, as long as the integration measure is real and positive as well.Were the symmetry spontaneously broken we should get for P (q) either a cosine function inthe simplest case, or a sum of cosines in the most general case. This sum takes positive andnegative values, but since the negative values are excluded, P (q) should be a constant functionequal to 1, representing a symmetric vacuum state.

Let us now extend, as much as possible, the Vafa-Witten result for pure gluonic operatorsto fermion bilinear local operators. To this end we will use the generalization of the p.d.f.discussed at the beginning of this chapter, which applies to local operators constructed withGrassmann fields. The main goal of the following lines is to show that all local bilinear P = −1gauge invariant operators of the form O = ψOψ, with O a constant matrix with Dirac, colorand flavour indices, take a vanishing vacuum expectation value in any vector-like theory withNF degenerate flavours.

Let us start with the one-flavour case since, as it will be shown, it is a special case. Thestandard hermitian, local and gauge invariant P order parameter bilinear in the fermion fieldsis

ψOψ = iψγ5ψ.

Equation (2.6) gives the generation function of all the moments of O

P (q) =

⟨

det(D + m + q

V γ5

)

det (D + m)

⟩

(4.24)

The determinant of the Dirac operator in the denominator of (4.24) is positive definite, butthe numerator of this expression, even if real, has not well defined sign. The final form for P (q)will depend crucially on the distribution of the real eigenvalues of γ5 (D + m). Therefore wecannot say a priori whether P (q) will be the constant function P (q) = 1 (symmetric vacuum)or any other function (spontaneously broken P ). The matter will be solved as we go on withthe N -flavoured case.

2As the key of the proof is the pure imaginary nature of the euclidean version of the external source Y (Aaµ) =

iX(Aaµ), Cohen’s remark still applies to the p.d.f. derivation of the Vafa-Witten Parity theorem.

52

For NF flavours (NF > 1), the most general P = −1 hermitian and Lorentz and gaugeinvariant local order parameters ψOψ that can be constructed are

iψγ5ψ, iψγ5τψ; (4.25)

with τ any of the hermitian generators of the SU(NF ) Flavour group. However, since Flavoursymmetry cannot be spontaneously broken in a vector-like theory [60], particularly in QCD3,we will restrict our analysis to the flavour singlet case, which should give us all the informationabout the realization of Parity

iψγ5ψ = iψuγ5ψu + iψdγ5ψd + iψsγ5ψs + . . . (4.26)

Let us assume that 〈iψuγ5ψu〉 = ±c0 6= 0. Since Flavour symmetry is not spontaneouslybroken,

〈iψuγ5ψu〉 = 〈iψdγ5ψd〉 = 〈iψsγ5ψs〉 = . . . (4.27)

Thus the system will show two degenerate vacua with all the condensates oriented in the samedirection. This ‘ferromagnetic’ behaviour is naturally imposed by the realization of Flavoursymmetry in the vacuum. Otherwise one could imagine also ‘antiferromagnetic’ vacua withantiparallel condensates, or even more complex structures.

The interaction between different flavours in vector-like theories is mediated by the particlesassociated to the gauge fields, gluons in QCD. Assuming 〈iψuγ5ψu〉 6= 0, the fact the Flavoursymmetry is conserved [60] suggests that the gauge interaction favours ‘ferromagnetic’ vacua,with parallel oriented condensates. However the actual dynamics can become more complicated.Indeed, a non vanishing condensate 〈iψuγ5ψu〉 6= 0, which would imply spontaneous breakingof P , CP , T and CT , can be excluded. Let us see how in the following lines:

If we apply equation (2.6) to the computation of the p.d.f. in momentum space Pud(q) ofiψuγ5ψu + iψdγ5ψd, the result is

Pud(q) =

⟨(

det(D + m − q

V γ5

)

det (D + m)

)2⟩

, (4.28)

where (D + m) in (4.28) is the one flavour Dirac operator and the mean value is computed inthe theory with NF -degenerate flavours.

The r.h.s. of equation (4.28) is the mean value, computed with a positive definite integrationmeasure, of a real non-negative quantity. Thus Pud (q) is positive definite, or at least a nonnegative, definite quantity. Were 〈iψuγ5ψu〉 non-zero, one should expect a cosine function forPud (q), since iψuγ5ψu and iψdγ5ψd are enforced to take the same v.e.v. because of Flavoursymmetry. Since the positivity of Pud(q) excludes such a possibility, we can infer that all thepseudo-scalar condensates iψfγ5ψf take vanishing expectation values.

The aforementioned considerations indeed complete Vafa and Witten’s theorem on Parityconservation on vector-like theories with NF degenerated flavours4. Nevertheless, in both theo-rems by Vafa and Witten, no regularisation mechanism is applied to the theory. The omissionof the regularisation step in what aims to be a general proof of symmetry conservation is a dan-gerous business, and as we shall see, can lead to inconsistencies when a particular regularisationis chosen.

3QCD with Wilson fermions is not an exception, for as we will see, it does not comply with the requirementsof the theorem [60].

4Except for the finite temperature issue, which will not be addresses in this dissertation.

53

One steps further: Parity and Flavour conservation in QCD

The last result is not as useful as one would like: It is not valid for NF non-degenerated flavours,which is the case of QCD at the physical point. Regarding the paper of Vafa and Witten [60]on vector-like symmetries, such as Flavour or baryon number conservation, it does not makereferences to any particular regularisation either, but this does not imply that the arguments arevalid for all the different versions of fermions in the lattice. In fact, the only two regularizationscapable of simulating just one quark flavour5, namely Wilson fermions and Ginsparg-Wilsonfermions, do not comply with the requirements of the theorem. In the Ginsparg-Wilson case,the theorem does not apply because, even if the integration measure is positive definite, theother essential ingredient in the proof in [60], the anticommutation of the Dirac operator withγ5, is not realized. For the case of Wilson fermions neither of the two assumptions in [60],positivity of the integration measure and anticommutation of the Dirac operator with γ5, arefulfilled, the first of the two failing for an odd number of flavours. Indeed there exists a regionof the parameters space where Parity and Flavour symmetries are spontaneously broken: thewell known Aoki phase [25, 26], and even a more complex phase structure for lattice QCD withWilson fermions has been suggested [51, 43], as discussed in the previous chapter. In the end,a theoretical proof of the realization of symmetries of QCD is still lacking.

The standard wisdom is that Vafa and Witten theorems fail, when applied to Wilsonfermions, due to the existence of exceptional configurations which have a non-vanishing weight inthe Aoki phase. Outside this phase, and in particular in the physical region near the continuumlimit, the exceptional configurations would be suppressed and then Parity and Flavour sym-metries would be restored in the QCD vacuum. Following this wisdom, and in order to proveParity and Flavour conservation in QCD [68], it would be very convenient to choose a ’smalleigenvalue free’ regularization for the fermions. It happens that Ginsparg-Wilson fermions fulfillthis requirement.

In order to follow the steps of the proof, some knowledge on the spectrum of the Ginsparg-Wilson operator is needed. This knowledge is exposed in appendix D. Here I restrict thecomputations to a particular realization of the Ginsparg-Wilson relation (R = 1

2 , see appendixD) verifying

{γ5, D} = aDγ5D, (4.29)

which is equivalent to overlap fermions. I also use as the mass and pseudoscalar terms theunsubstracted order parameters

iψ

(

1 − aD

2

)

ψ iψγ5

(

1 − aD

2

)

ψ, (4.30)

which transform well under chiral rotations. Finally, the Ginsparg-Wilson operator ∆ invokedin the following pages refers to

∆ =(

1 − am

2

)

D, (4.31)

for the reasons explained in appendix D.

5There is a polemical controversy around the staggered quarks with the rooting technique. One side arguesthat they should be able to simulate one flavour and give the right continuum limit [12], and the other explainsthat some features of the theory in question can not be reproduced by the rooting [14]. The analysis of thisdiscussion is out of the scope of this dissertation.

54

The one flavour case

Let us start by analyzing the one flavour case with the p.d.f.. As before, we work with theP -violating bilinear O = iψγ5ψ. Our basic tool in the analysis is P (q), the generating functionfor the moments of O,

P (0) = 1, P ′ (0) = i 〈O〉 ,P ′′ (0) = −

⟨O2

⟩. . . Pn (0) = (i)n 〈On〉 .

(4.32)

The evaluation of the generating function is straightforward using (4.24)

P (q) =

⟨

det(∆ + m + q

V γ5

)

det (∆ + m)

⟩

=

⟨

det(H + q

V

)

det H

⟩

, (4.33)

where H = γ5 (∆ + m). Let us write the P (q) in terms of the µj , the eigenvalues of H

P (q) =

⟨∏

j

(µj + q

V

)

∏

j µj

⟩

=

⟨∏

j

(

1 +q

µjV

)⟩

. (4.34)

In appendix D it is shown that the modulus of the eigenvalues |µj | is bounded from below bythe mass, |µj | ≥ m (see equation (E.22)). Thence, no zero modes are allowed if m > 0, andevery factor of the rightmost member of equation (4.34) is well defined and does not diverge.The expansion of the product of the eigenvalues leads to a new expression,

P (q) =V∑

k=0

qk

⟨

1

V k

∑

(j1,··· ,jk)

1

µj1 , · · · , µjk

⟩

, (4.35)

where the sum over (j1, · · · , jk) means that every possible combination of the indices j1, · · · , jk

is added to the final result. This sum can be arranged in a simpler fashion up to finite volumeeffects,

1

V k

∑

(j1,··· ,jk)

1

µj1 , · · · , µjk

=1

k!

1

V

∑

j

1

µj

k

+ O

(1

V

)

. (4.36)

As the results derived here are only valid in the thermodynamic limit, the residual terms oforder O

(1V

)should not be a problem.

The analysis of the spectrum of the Ginsparg-Wilson operator (see appendix D) reveals thatthe µ’s come in pairs ±µ, except for the ones corresponding to chiral modes, in case they exist.Therefore, most of the terms in (4.36) cancel out, and the only contribution left is the onecoming from the chiral modes,

1

V

∑

j

1

µj=

1

V

(1

m

(n+ − n−)

+a

2

(n′+ − n′−)

)

=

(am

2− 1

) Q

mV, (4.37)

where n± is the number of zero modes with chirality ±1, whereas n′± is the number of realmodes λ = 2

a with chirality ±1. Putting everything together, we conclude, for non-zero mass

P (q) =V∑

k=0

qk

k!

(am

2− 1

)k 1

mk

⟨(Q

V

)k⟩

+ O

(1

V

)

. (4.38)

55

This relationship implies that the moments of the p.d.f. of O = iψγ5ψ and those of the densityof topological charge are intimately related. The relationship is somewhat expected, as it isvery well known that for Ginsparg-Wilson fermions

χ5 = −⟨ψψ

⟩

m+

χT

m2.

The exact relation between the moments of these two magnitudes can be easily calculated. Forthe even moments, the result is

⟨(iψγ5ψ

)n⟩= (−i)n

(am

2− 1

)n 1

mn

〈Qn〉V n

+ O

(1

V

)

, (4.39)

and all the odd moments vanish by Parity. The first non-trivial moment is the second one

⟨(iψγ5ψ

)2⟩

= −(am

2− 1

)2 1

m2

⟨(Q

V

)2⟩

+ O

(1

V

)

. (4.40)

At this point we require that iψγ5ψ be an hermitian operator, in order to carry on with theproof. This is a simple requirement, but with important consequences: The expectation valueof the square of an hermitian operator must be positive, but from (4.40), this expectation valueis manifestly negative in the thermodynamic limit. The only way to fulfill both requirementsat the same time is the vanishing of the second moment,

limV →∞

⟨(Q

V

)2⟩

= 0, (4.41)

but then the probability distribution function of the density of topological charge, QV , must

become a Dirac delta at the origin as V increases,

limV →∞

p

(Q

V

)

= δ

(Q

V

)

, (4.42)

and all the higher moments of both iψγ5ψ and QV vanish as well. Therefore Parity is not broken

in lattice QCD with one flavour of Ginsparg-Wilson fermions, at least for the standard orderparameter iψγ5ψ.

Let us consider now the case of the unsubstracted order parameter,

iψγ5

(

1 − aD

2

)

ψ. (4.43)

Its generating function P (q) is

P (q) =

⟨

det[∆ + m + q

V γ5

(1 − aD

2

)]

det (∆ + m)

⟩

. (4.44)

The expression (4.44) is immediately simplified, for only the zero modes of D contribute to thislast equation (4.44). In order to see it, let us consider the following argument: The matrixcorresponding to the numerator is block-diagonal, whose blocks are of the form indicated in(E.24). The contribution to the ratio coming from a pair of complex eigenvalues of D belongingto a 2 × 2 block is of the form (from (E.25) in appendix D)

1 − αq2

V 2, (4.45)

56

with |α| ≤ m−2 (E.27). Therefore, the contribution to P (q) corresponding to complex eigen-

values comes solely from the factor∏

j

(

1 − αjq2

V 2

)

, where the product extends over all pairs

of complex eigenvalues. Expanding this product, the coefficient corresponding to q2k can becomputed easily,

∣∣∣∣∣∣

1

V 2k

∑

(j1,...,jk)

αj1 · · ·αjk

∣∣∣∣∣∣

≤ 1

V 2k

∑

j

|αj |

k

≤ V km−2k

V 2k=

m−2k

V k. (4.46)

Thence, the contribution from the complex eigenvalues is of order 1 + O(

1V

), that is, just 1 in

the thermodynamic limit.Regarding the chiral modes of D with λ = 2

a , and as explained in (E.28), they also contributewith a factor 1. The zero modes of D, on the other hand, give a non-trivial contribution,

P (q) =(

1 +q

mV

)n+ (

1 − q

mV

)n−

. (4.47)

If n+ < n− (Q > 0), then

P (q) =

(

1 − q2

m2V 2

)n+(

1 − q

mV

)Q. (4.48)

A similar expression is valid for n+ > n−. The argument applied to (4.46) can be essentiallytranslated here: The first factor of the r.h.s. of (4.48) goes to 1 in the thermodynamic limit.Therefore we obtain the final result (valid for arbitrary values of n+ and n−) when V → ∞

P (q) =(

1 − sign(Q)q

mV

)|Q|. (4.49)

All odd moments vanish as before because of (finite-volume) Parity symmetry. The even mo-ments also vanish because they are trivially related to the ones in (4.38), as can be seen easilyby expanding (4.49). In consequence, we see that Parity is not broken for the unsubstractedorder parameter either.

It is interesting to give a more physical argument that uses only the vanishing of the secondmoment. In fact, if Parity were spontaneously broken, we would expect two degenerate vacuaα and β, since Parity is a Z2 symmetry. Let zα be a complex number which give us the meanvalue of the pseudoscalar P in the α state

〈P〉α = zα,

then we have

〈P〉β = −zα.

Since P2 is Parity invariant, it takes the same mean value in the two states. Making use ofthe cluster property in each one of these two states, one can find out that

⟨P2

⟩=

1

2

⟨P2

⟩

α+

1

2

⟨P2

⟩

β= z2

α,

but since⟨P2

⟩= 0, zα must vanish.

In conclusion we have shown rigorously, assuming hermiticity of iψγ5ψ and using standardproperties of Ginsparg-Wilson fermions, that Parity is not spontaneously broken in the one-flavour model, at least for the more standard order parameters, namely iψγ5ψ and the densityof topological charge.

57

The NF flavours case

The last arguments apply only to the one-flavoured case, but they can be easily extended to anarbitrary number of flavours NF . In contrast with the extension of the Vafa-Witten theoremproposed before, these flavours can have different masses, but they must be non-vanishing.Most of the results from the previous section apply here as well with small modifications. Thefermionic action is

NF∑

α=1

ψα (∆ + mα)ψα. (4.50)

The complete spectrum of the Dirac operator consists of NF copies of the single flavour spec-trum, each of them calculated with the mass of the corresponding flavour.

Let us consider the usual pseudoscalar order parameter for a single flavour β, iψβγ5ψβ. Thecorresponding generating function P (q) can be computed easily,

P (q) =

⟨

det(∆ + mβ + q

V γ5

)

det (∆ + mβ)

⟩

NF

. (4.51)

The average is taken over the effective gauge theory with NF flavours, but only the β flavourappears within the average6. The calculation is identical to the one for the single flavour caseand gives

P (q) =

⟨∏

j

(

1 +1

µβj

q

V

)⟩

. (4.52)

The superindex on the eigenvalues indicate flavour, that is, µβ belongs to the spectrum ofγ5 (∆ + mβ). Equation (4.52) is essentially the same as the one for one flavour, and the momentsof P (q) still are, up to constants, the same as the moments of the density of topological chargeQV . The only difference is that now the average is taken in the theory with NF flavours. Thisdoes not change any of the conclusions: all odd moments vanish by symmetry, the secondmoment vanish in the thermodynamic limit because of the hermiticity of iψγ5ψ, the densityof topological charge goes to a Dirac delta centered on the origin, and therefore all the highermoments vanish as V → ∞. This is valid for any flavour separately, and so will be validas well for any linear combination

∑

α Aαiψαγ5ψα. The extension to the unsubstracted orderparameter is trivial. This result extends the generalization of the Parity Vafa-Witten theoremto fermionic bilienars, but note that the mass of every flavour must be non-zero, otherwise theproof breaks down.

Let us consider now the degenerate case, all flavours with equal nonzero masses. The actionnow enjoys Flavour symmetry. As stated in the introduction of this paper, the Vafa-Wittentheorem [60] for vector-like symmetries does not apply to Ginsparg-Wilson fermions because,even if the integration measure is positive definite, the Dirac operator does not anticommutewith γ5. However we can study this symmetry with the p.d.f. method as we did for Parity.

Consider first the case of two degenerate flavours and the standard order parameters ψτ3ψand iψγ5τ3ψ. Proceeding as before, we find for the first order parameter

P (q) =

⟨

det(∆ + m + i q

V

)det

(∆ + m − i q

V

)

det (∆ + m)2

⟩

NF

=

6Our fermionic determinant (and hence, its eigenvalues) will always refer to a single flavour.

58

⟨∏

j

(

1 +q2

λ2jV

2

)⟩

NF

, (4.53)

where λj are the eigenvalues of ∆ + m. By the same argument we have use repeatedly before,i.e., the lower bound of the eigenvalues |λ| < 1

m , P (q) → 1 in the thermodynamic limit7.

Similarly for iψγ5τ3ψ we obtain

P (q) =

⟨

det(H + q

V

)det

(H − q

V

)

det (H2)

⟩

NF

=

⟨∏

j

(

1 − q2

µ2jV

2

)⟩

NF

. (4.54)

The same argument applies, for |µ| is bounded from below as well, and we have also thatP (q) → 1 as the volume grows to infinity. In fact, the extension of the proof of Flavourconservation to NF flavours is trivial: since the preceding results apply to any pair of flavours,and we can choose the generators τi of the SU (NF ) Flavour symmetry group so as to involveonly pairs of flavours, then Flavour symmetry is realized in general. The calculations canbe repeated easily for the unsubstracted operators, and the result is the same. We can thenconclude that there is no Aoki phase for two flavours of Ginsparg-Wilson fermions with non-zero mass. This conclusion should not be surprising at all, for the spectrum of the hermitianGinsparg-Wilson operator H with a mass term is depleted of small eigenvalues of order ≈ 1

V .

The following relation

χ5 = −⟨ψψ

⟩

m+ N2

F

χT

m2,

also holds when taking into account the number of flavours, implicitly in the condensate, andexplicitly in the topological susceptibility via the N2

F factor.

The chiral limit

The results of the previous sections can not be extended in a straightforward way to QCD inthe chiral limit. We lose the non-trivial lower bound on the spectrum of ∆ + m, 0 < m < |λ|,and the Dirac operator has exact zero modes corresponding to gauge fields with non-trivialtopology. Thence, as the arguments developed in the preceeding sections do not help us findinga definite conclusion on the realization of Parity in the chiral limit from first principles, we willappeal to the standard wisdom.

Let us consider QCD with one massless flavour. In this limit, the action of the model hasthe chiral U(1) symmetry, which is anomalous because the integration measure is not invariantunder chiral U(1) global transformations. The Jacobian associated to this change of variablesintroduce an extra term to the pure gauge action proportional to the topological charge ofthe gauge configuration, the θ-vacuum term, which allows one to understand the absence of aGoldstone boson in the model, but this fact generates the well known strong CP problem. Allthese features of QCD in the continuum formulation are well reproduced in lattice QCD withGinsparg-Wilson fermions, as discussed in this chapter and appendix E.

7This conclusion rests only on the bound on the eigenvalues, and not on any other specific property of theDirac operator.

59

Let’s consider first the unsubstracted scalar order parameter, ψ(1 − aD

2

)ψ. The correspond-

ing generating function P (q) in the chiral limit and for a generic value of θ8 is

P (q) =

∫[dA] e−SPGe−iθQ det

(D + i q

V

(1 − aD

2

))

Z, (4.55)

with Z =∫

[dA] e−SPGe−iθQ detD and Q = n− − n+. The contribution to the determinant inthe numerator of (4.55) coming from pairs of complex eigenvalues of D is easily computed andgives a factor

f0 (q) =∏

j

[

|λj |2 −q2

V 2

(

1 − a2|λj |24

)]

, (4.56)

where the product is taken over all different pairs of complex eigenvalues. Each chiral modecorresponding to an eigenvalue 2

a contributes a factor of 2a to the determinant, and each chiral

mode corresponding to a zero eigenvalue contributes a factor of iqV . The normalization factor Z

is computed from the same expressions by setting q = 0. Therefore we can write P (q) as

P (q) = Z−1

∫

[dA] e−SPGe−iθQf0(q)

(2

a

)n′++n′− (iq

V

)n++n−

, (4.57)

where n± represent the chiral zero modes, and n′± represent the chiral 2a modes. The compu-

tation of the generating function for the pseudoscalar order parameter iψγ5

(1 − aD

2

)ψ follows

the same line,

P (q) = Z−1

∫

[dA] e−SPGe−iθQf0(q)(−1)n+

(2

a

)n′++n′− ( q

V

)n++n−

. (4.58)

Defining

fS (q) = f0 (q)

(2

a

)n′++n′− (iq

V

)n++n−

(4.59)

and denoting by PS (q) and PP (q) the generating functions for the scalar and pseudoscalarrespectively, we can rewrite the above results in the following way

PS (q) = Z−1S

∫

[dA] e−SPGe−iθQfS (q) , (4.60)

PP (q) = Z−1P

∫

[dA] e−SPGe−iθQfS (q) (−i)Q . (4.61)

Two comments are in order, regarding these two expressions:

(i.) Only the Q = ±1 sectors verifying n− + n+ = 1 contribute to the chiral condensate 〈S〉.As its expectation value is computed as the derivative dPS

dq

∣∣∣q=0

, only when n− + n+ = 1

all the q factors are removed from the expression (4.60), giving a non-zero result.

(ii.) The dependence of the expectation value of the scalar order parameter with the θ param-eter is 〈S〉 ∝ cos θ.

8Although in the chiral limit the free energy does not depend on θ, the condensate certainly does, as in orderto compute it in the chiral limit we need to know the free energy in the neighbourhood of the point m = 0. Thisneighbourhood is θ dependent.

60

For the second moment⟨S2

⟩, the only non-vanishing contributions come from the sectors with

Q = 0 and Q = ±29, the first coming from the second derivative of the function f0 (q), and the

second appearing as the factor(

iqV

)2is eliminated by the second derivative. Thence

⟨S2

⟩= A0 + A2, (4.62)

where A0 is the contribution coming from the Q = 0 sector and A2 the contribution comingfrom the Q = ±2 sector.

Expression (4.61) for the pseudoscalar condensate tells us that all the odd moments vanishbecause of Parity symmetry10. In contrast, the second moment receives non-zero contributionsfrom the Q = 0,±2 sectors

⟨P 2

⟩= A0 − A2. (4.63)

Looking at moments of higher order we would find an infinite set of relations among the Ai.

Since strictly speaking, due to the chiral anomaly we do not have a new symmetry in thechiral limit of one flavour QCD, the standard wisdom tells us that the vacuum expectation valueof the chiral order parameter does not vanish as m → 0. Moreover, the absence of masslessparticles in the one flavour model suggests that the perturbation series in powers of m doesnot give rise to infrared divergences [46], the free energy density is an ordinary Taylor series inm [69, 46]; and in what concerns the chiral condensate, the chiral and thermodynamical limitscommute.

On the other hand, the free energy density of the model at m 6= 0 and θ = 0 can be computedin the thermodynamical limit from the topologically trivial sector Q = 0 [69, 46]. But sincechiral symmetry in the Q = 0 sector is not anomalous, it should be spontaneously broken inthat sector, if the chiral condensate takes a non vanishing value when approaching the chirallimit. In such a case, the value of the chiral condensate in the full theory and in the chiral limitwill be related to the spectral density at the origin of eigenvalues of the Dirac operator of thetopologically trivial sector by the well known Banks and Casher formula [44]

〈S〉 = −πρ(0) = Σ0.

This equation provide us with a non trivial relation between the value of the scalar condensatein the chiral limit in the full theory, which gets all its contribution from the Q = ±1 sectors,and the spectral density of eigenvalues at the origin of the Dirac operator in the topologicallytrivial sector Q = 0.

The scalar condensate is invariant under Parity, and therefore we expect its probabilitydistribution function to be a delta function δ(c − Σ0) in the full theory in the chiral limit,irrespective of the realization of Parity in the vacuum. Thence, we expect for the moments

〈Sn〉 = Σn0 . (4.64)

For the second moment, this implies

A0 + A2 = Σ20, (4.65)

9As in the case of 〈S〉, only when Q = ±2 and n−+n+ = 2 there appears a contribution to the second momentof the pseudoscalar. This rule holds for every contribution coming from a sector with non-trivial topology Q 6= 0.

10Parity symmetry imposes that, given a configuration with topological charge Q = n, one always can find acorresponding configuration with topological charge Q = −n, and the same weight in the partition function. Asthe factor (−i)Q changes sign with Q when Q is odd, the sum of the contributions of every pair of correspondingconfigurations vanishes for any observable. Even values of Q do not share this property, and (−i)2n = (−i)−2n.

61

but as previously stated, the standard wisdom suggests that the topologically trivial sectorbreak spontaneously the chiral U(1) symmetry, and since A0 is exactly the second moment ofPS(c) computed in the Q = 0 sector, what should happen is

A0 =1

2Σ2

0, (4.66)

and consequently

A2 =1

2Σ2

0, (4.67)

but this implies that the second moment of the pseudoscalar condensate (4.63) vanishes,

⟨P2

⟩= 0. (4.68)

By a similar argument, the relations for the higher (even) moments,⟨S2n

⟩= Σ2n

0 imply thevanishing of the corresponding pseudoscalar moments,

⟨P2n

⟩= 0

Symmetry under Parity is the only obvious reason for the vanishing of the pseudoscalarmoments, and therefore the previous result strongly suggests that Parity is also realized inQCD with one massless flavour.

4.2 Conclusion

It is quite well-establish in the scientific community that Parity and vector-like global symmetriesremain unbroken in QCD. Nonetheless, no sound theoretical proof of this hypothesis has everbeen presented. Vafa and Witten gave convincing arguments against spontaneous breaking ofthese symmetries in [59, 60], but those were questioned by several groups [32, 63, 61, 62, 65, 66],and now the QCD researchers agree on the lack of a proof for Parity realization in QCD. As faras vector-like symmetries –as Flavour or Baryon number conservation- are concerned, it mustbe remarked that the staggered fermion discretization is the only known lattice regularizationthat fulfills the initial conditions of the Vafa-Witten theorem. Indeed, the fact that the theoremis not applicable neither to the Ginsparg-Wilson regularization nor to Wilson’s one is quiteremarkable. Both regularizations violate the condition of the theorem that the Dirac operatormust anticommute with γ5. Moreover, in the Wilson case positivity for the integration measurecannot be assumed for a single flavour. As a matter of fact, the Wilson regularization featuresthe Aoki phase, where Parity and Flavour symmetries are spontaneously broken.

The p.d.f. formalism, when combined with an appropiate regularization, shows a way todiscard spontaneous breaking of these symmetries in QCD. By an appropiate regularization wemean a regularization where the eigenvalues of the Dirac operator are bounded from below fora non-zero fermion mass. This is not the Wilson case, and those arbitrarily small eigenvaluesare responsible of the appearance of the Parity and Flavour breaking Aoki phase. Fortunately,the Ginsparg-Wilson regularization has nice chiral properties which protect the masses of thequarks to be renormalized additively. This is translated into our desired lower bound for theeigenvalues, and thence, a proof for Parity and flavour conservation can be constructed. Thisis a major result that overcomes the difficulties found in [59, 60].

62

Chapter 5

The Ising model under an imaginarymagnetic field

“The distance between insanity andgenius is measured only by success.”

—Bruce Feirstein

5.1 Introduction to the Ising model

The origins of the Ising model

The Ising model was first proposed1 by Lenz to his student Ising [72] as a simple model describingferromagnets. Lenz’s original model tried to explain the Curie law of paramagnets [73], evadingthe widely criticized molecular-field hypothesis of Weiss [74]. Although Weiss hypothesis wassuccessful numerically, and in fact, when combined with the Langevin equations [75], was ableto predict the Curie law and the phenomenon of spontaneous magnetization, it awoke sharpcriticism among many physicists. The most remarkable of them was Stern (1920), who arguedthat Langevin equations were not applicable to gases at very low temperatures, in a crystalinesolid state. The gas molecules carrying magnetic moment were implicitly suppossed to rotatefreely, but this assumption could not be extended to the crystaline state. Stern attributed thesuccess of the Weiss hypothesis to its many adjustable parameters.

Free rotability was in clear contradiction with Born’s theory of crystal structure, yet itseemed an indispensable ingredient to obtain the Curie law and spontaneous magnetization.Lenz tried to solve the contradiction using some sort of quantum approach (quantum in thevery old sense of the first years of quantum mechanics). He assumed that the gas moleculeswere unable to rotate freely inside the crystal, however, they could perform turnover movements,and the positions allowed for each molecule were effectively quantized. He argued carefully thatin the end, there were only two distinct positions available for these molecules.

In the case these molecules did not interact, Lenz found the Curie law for paramagnets.Lenz expected the interacting case to lead to spontaneous magnetization, nonetheless he nevermade computations on the interacting case, or specified any particular interaction among themolecules. He assigned this task to his student Ising.

Ising’s intuition was the posibility of generating a non-local effect associated to spontaneousmagnetization from a local nearest-neighbour interaction. He expected the forces acting between

1Interesting reviews of the Ising model can be found in [70, 71].

63

molecules to be short ranged, and deduced that qualitative physics could be extracted froma nearest-neighbour approximation. Ising solved the one-dimensional version of the model[76], the chain model, and found no signal of spontaneous magnetization. He obtained thesame result in some variations of the model, including an arrangement of several chains inthree dimensions; thence he incorrectly assumed that the three-dimensional case displayed nospontaneous magnetization.

Ising result was in some way dissappointing. People hoped that the main properties offerromagnets could be described by such a simple model, and after the failure, they almostforsook it. It took some years for the scientific community to regain the faith in the Isingmodel2, and the main responsible was Onsager, who solved analitically the two-dimensionalcase, and showed how spontaneous magnetization appeared below a critical temperature [80].From that point on, the number of papers on the Ising model increased notably.

Onsager’s solution is usually regarded as brilliant and extremely complicated. Indeed, upto this moment, there is no known solution for higher dimensional versions of the Ising model(except for the infinite-coupled model). Even the two-dimensional case in the presence of amagnetic field remains unbeaten.

Why the Ising model?

The Ising model is a good starting point to test our θ vacuum algorithms for several reasons.First of all, it is easy to treat and code into a computer, and the simulations are fast enoughto allow us to generate large statistics, even in large lattices. Furthermore, the one-dimensionalmodel in the presence of a magnetic field is exactly solvable, so we can check our results againstanalytic formulae in order to make sure they work. Then, the application of our methodsto higher dimensions is straighforward. On the other hand, we can identify in some sensemagnetization and topological charge on this model, and regard an imaginary external magneticfield as a θ term in the action. Finally, from the numerical point of view, it is even morechallenging3 than other complex systems suffering from the sign problem, as lattice QCD witha θ−term, yet it remains more accessible. Therefore, it is a good idea to check the goodness ofany algorithm in these toy models, prior to their application to more complex systems. Fromthe pedagogical point of view, the Ising model as an introduction to the lattice is also a goodstart.

5.2 Solving the Ising model

Definition of the Ising model

The Ising model is an statistical system described by the following hamiltonian

H ({si}, J, B) = −J∑

{i,j}sisj − B

∑

i

si, (5.1)

where the si are Z2 variables (spins4 from now on) in the {1,−1} representation, placed on aregular lattice, and {i, j} means sum over nearest neighbours. J is the coupling constant amongspins and B stands for the external magnetic field.

2See for instance [77, 78, 79].3As the following sections show, the phase diagram of the Ising model within an imaginary magnetic field is

richer than the one expected for QCD in presence of a θ–vacuum term.4The spin nomenclature appeared a long time after the model was devised, probably linked to the apparition of

the Heisenberg model. At that time, the Ising model became to be regarded as a simplification of the Heisenbergmodel.

64

As I said, this is an statistical system. By an statistical system I mean a system whosedynamics are completely irrelevant. We do not know the equations of motion of this model,and in fact, we do not need to. The kinetic term is even absent in the hamiltonian. We treatthese systems by generating ensembles of configurations and then performing statistical analysis,which involves the computation of mean values and correlation functions. A configuration is aset of allowed values for the spin variables si. The usual way to represent a configuration is byassigning arrows (↑= +1 and ↓= −1) to each value of the spin, and put one arrow per latticesite:

↑↑↓↑↑↑↓↓↑↓↓↑↓↓↑↓↑↑↓↓↓↑↓↓↓↑↓↓↑↑↑↓↑↓↓↑↓↓↑↓↑↑↑↓↑↑↑↓↑↓↓↓↓↓↓

The energy of each configuration is computed by using (5.1). It is obvious from (5.1) that, atzero magnetic field, the operation si → −si keeps the energy of the system unchanged. Thisfact is a consequence of the Z2 symmetry the system enjoys in the case of vanishing magneticfield. The quantity

M =N∑

i

si (5.2)

is not invariant under Z2 transformations, and defines an order parameter of the Z2 symmetry.This quantity is called the total magnetization of the system, so for symmetry reasons we naivelyexpect M to be zero in the thermodynamic limit. The total magnetization may be understoodas a topological charge in those systems with an even number of spins5, in the sense that M/2is a quantized number, ranging from −∞ to ∞ in the thermodynamic limit. We will use thisproperty to analyze the Ising model as a toy model to test algorithms to simulate θ-vacuumtheories.

Direct application of Boltzmann hypothesis gives us the probability P ({si}) of each config-uration {si},

P ({si}) = e− H

kBT

Z = eF

P

{i,j} sisj+ h2

P

i si

Z

Z (F, h) =∑

{si} eFP

{i,j} sisj+h2

P

i si

, (5.3)

and in the case of zero external field h = 0, the Z2 symmetry forces explicitly the equalityP ({si}) = P ({−si}). The quantity Z is the partition function, which encodes all the informa-tion of the system6. The expectation value of any observable can be computed, once we know Z.As we have marked in the last equation, Z is a function of the system parameters, F = J/kBTand h = 2B/kBT , which I will call reduced coupling and reduced magnetic field. Thence, theaverage values of any observable or correlation function depend on these two parameters as well.

For instance, the average density of magnetization 〈m〉 = 1N 〈∑N

i si〉 is

5This requirement is a natural imposition in antiferromagnetic systems, if we want to avoid frustration.6To compute certain observables, however, it might be necessary to calculate the partition function coupled to

certain external sources. For instance, in order to compute the correlation functions a site dependent magneticfield Bi is required, as we will see in the following pages.

65

〈m〉 =∑

{si}P ({si})m ({si}) =

∑

{si}m ({si})

e− H

kBT

Z=

∂Z∂ h

2

Z. (5.4)

At a finite volume, the partition funcion can not describe a phase transition of any kind. Theproof is quite simple: The exponential7 e−βH is a positive analitic function of the Hamiltonianparameters J and B, and of the temperature β; as Z is a finite sum of exponentials, it is a non-zero analitical function as well. No singularities occurr in the computation of any observable,and thence, no phase transition behaviour is displayed.

In order for the phase transition to appear, we need to reach the thermodynamic limit. Then,the partition function becomes a serie, and singular behaviour may happen. Unfortunately, thepartition function is not well defined in the thermodynamic limit: When N → ∞, Z usuallydiverges8. Therefore, it is convenient to work with the free energy

f (F, h) = limN→∞

1

Nln (Z (F, h)) . (5.5)

No information is lost from Z (F, h) to f (F, h), thus we expect to be able to work out anyexpectation value from the free energy. In fact, taking derivatives with respect to the reducedmagnetic field we find the mean magnetization density of the system,

〈m〉 =∂f (F, h)

∂ h2

=1

N

∂ ln (Z (F, h))

∂ h2

=1

N

∂Z∂ h

2

Z. (5.6)

The response of the mean magnetization density to the reduced magnetic field is the definitionof the magnetic susceptibility

χ =∂〈m〉∂ h

2

=1

N

∂2Z

∂(h2 )

2

Z2−

∂Z∂ h

2

Z

2

=

N[⟨

m2⟩− 〈m〉2

]

= N (∆m)2 . (5.7)

The equality (5.7) is called fluctuation-dissipation theorem. It relates the fluctuations of thesystem with the sensitivity of the system to external fluctuations. The meaning is clear: Ahigh sensitivity to external agents also implies high sensitivity to thermal fluctuations, hencea system that shows a great response under external fields fluctuates wildly in the absence ofthese fields, and vice-versa.

The magnetic susceptibility is inherently related to the spin-spin correlation functions. Thesefunctions measure the influence of the spin j over the spin k. In order to compute these spin-spincorrelations, we introduce a site-dependent magnetic field

H ({si}, J, Bi) = −J∑

{i,j}sisj −

∑

i

Bisi. (5.8)

Now, the derivative of the free energy with respect to Bj yields the mean of the spin variableat the site j,

7As usual, β = 1

kBT.

8If a sign problem exists, the partition function becomes zero in the thermodynamic limit.

66

〈sj〉 =∑

{si}P ({si}) sj =

∑

{si}sj

eH

kBT

Z=

∂Z

∂hj2

Z=

∂f

∂hj

2

. (5.9)

It is straightforward to see that the taking of a second derivative with respect to the reducedmagnetic field in another site produces the desired correlation function

〈sjsk〉 =∑

{si}P ({si}) sjsk =

∑

{si}sjsk

eH

kBT

Z=

∂2Z

∂hj2

∂hk2

Z. (5.10)

For this reason, the partition function is called the generator of the correlation functions.

This quantity measures effectively the influence of sj in sk whenever 〈sj〉 = 〈sk〉 = 0. How-ever, if 〈sj,k〉 6= 0, the correlation function may return non-zero values, even if the fluctuationsof the spins sj and sk are completely unrelated. That is why we are usually more interested inthe connected correlation funcion

Gjk = 〈sjsk〉 − 〈sj〉〈sk〉, (5.11)

for this quantity measures the relations between the fluctuations of any pair of spins in anycase. An easy way to compute Gjk is to use the free energy, instead of the partition function

Gjk = 〈sjsk〉 − 〈sj〉〈sk〉 =∂2f

∂hj

2 ∂ hk

2

, (5.12)

and the free energy is the generator of the connected correlation functions.

Comparing (5.7) and (5.12), we can relate again the fluctuations of the spins with theresponse to a magnetic field

χ =∑

j,k

Gjk, (5.13)

and the fluctuation-dissipation theorem adquires its full meaning.

We qualitatively expect (5.11) and (5.12) to vanish as the distance between site j and kgrows: If sites j and k are far away from each other, it is reasonable to think that the spins atthe neighbourhood of site j are not affected by the fluctuations of the very distant spins of sitek. We should expect then

lim|j−k|→∞Gjk = 〈sjsk〉 − 〈sj〉〈sk〉 = 0, (5.14)

so, in this limit,

〈sjsk〉 = 〈sj〉〈sk〉, (5.15)

and applying translation invariance,

χ ≈ N∑

k

Gjk, (5.16)

for Gjk should depend only on |j − k|, and not on the particular site j we are sitting.

Equation (5.15) profiles what is called the cluster property, and enables us to define pureand mixed states in our system.

67

The cluster property for the Ising model

Let us assume that at a given temperature, the system is in a disordered phase9. If we draw anhistogram of the magnetization density, we observe something quite similar to figure 5.1.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

P(m

)

Magnetization m

Figure 5.1: Magnetization density distribution function in the disordered phase of the finitevolume Ising model.

As the number of spins increases, the width of the distribution narrows, until it becomesa Dirac delta. The position of the Dirac delta is uniquely determined by the intensity of themagnetic field. The important point to be noticed here is the fact that

〈mn〉 = 〈m〉n, (5.17)

since the probability distribution function of the magnetization density is a Dirac delta

∫

mnδ (m − m0) = mn0 . (5.18)

This is the cluster property, and as a consequence of it, wherever this property holds, theintensive magnitudes of the system do not fluctuate in the thermodynamic limit. The system issaid to be in a pure state.

This is not in contradiction to a non-zero response of the system to a magnetic field. Re-calling (5.7)

∂〈m〉∂ h

2

= N[⟨

m2⟩− 〈m〉2

]

= N (∆m)2 , (5.19)

we observe that the vanishing of 〈m2〉−〈m〉2 is competing against the divergence N → ∞. Thisclash of magnitudes results in a finite, non-zero value for the magnetic susceptibility.

This analysis is correct for the disordered phase. A breakdown of the Z2 symmetry of thesystem (which may or may not happen at a given temperature) changes this picture drastically.The distribution function of the magnetization density now features two identical symmetricpeaks (see fig. 5.2).

9By disordered phase I understand that no spontaneous Z2 symmetry breaking has occurred, so 〈si〉 = 0, thespins point up or down randomly, and the correlation functions die at long distances.

68

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

m− m+-1.00 -0.50 0.00 0.50 1.00

P(m

)

Magnetization m

Figure 5.2: Magnetization density distribution function in the ordered (magnetized) phase ofthe finite volume Ising model.

The configuration space is divided, and this division becomes disconnected in the thermody-namic limit. Thus, if the system lies in the region of magnetization density m+, it will remainthere forever, never crossing to the region of configurations with magnetization density m−.The connection only happens at a finite volume; then the transitions between these two distinctregions of the phase space are allowed. The effect of a magnetic field here is the selection of thefavoured peak.

As we repeat the calculation of the equation (5.18), we discover that the cluster property isviolated here

∫

mn 1

2(δ (m + m−) + δ (m − m+)) =

mn+ + mn

−2

, (5.20)

for we have a set of vacua, as a consequence of the spontaneous Z2 symmetry breaking. The fluc-tuation from one vacuum to another spoils the cluster property, and the susceptibility diverges.The system lies in a mixed state.

These kind of fluctuations only happen at the critical point. In the broken phase, as we reachthe thermodynamic limit, the system is always in one of the two possible pure states10 . On theother hand, for a finite volume, the transition can always take place, and the cluster property isnot rigorously fulfilled; obviously, the cluster property only makes sense in the thermodynamiclimit.

Analytic solution of the 1D Ising model

The one-dimensional case of the Ising model is exactly solvable, as Ernst Ising himself demon-strated in his PhD thesis in 1925. There are many ways to compute the free energy of the Isingmodel; the one we show here makes use of the transfer-matrix formalism, to illustrate how thistechnique can be applied to solve simple 1D models.

The clearest way to apply the transfer-matrix method to the one-dimensional Ising modelrequires a superficial modification of the Hamiltonian

10For a continuous symmetry, the Goldstone modes enable the system to change the vacuum in the thermody-namic limit as well. Then, both the susceptibility and the correlation length diverge in the broken phase.

69

H ({si}, J, B) = −JN∑

i

sisi+1 −B

2

N∑

i

(si + si+1) , (5.21)

where I am taking two assumptions for granted: First, there exist some kind of spin ordering,so as to the spin si is a neighbour in the one-dimensional spin-chain of the spins si−1 and si+1.Second, there are periodic boundary conditions, and the spin sN+1 is the same as the spin s1.Then this Hamiltonian is completely equivalent to (5.1) with periodic boundary conditions.

Now we can decompose (5.21) in small site-dependent functions

H ({si}, J, B) =N∑

i

I (si, si+1, J, B) , (5.22)

where

I (si, si+1, J, B) = −Jsisi+1 −B

2(si + si+1) . (5.23)

After these modifications, the computation of the partition function can be easily done. Weapply the definition

Z (F, h) =∑

{si}e−βH(J,B) =

∑

{si}e−

PNi βI(si,si+1,J,B)

=∑

{si}

N∏

i

e−βI(si,si+1,J,B) = tr(TN

). (5.24)

where the matrix T is called the transfer-matrix. It is defined as a matrix of possible states, inthis case restricted to a two spin configuration by construction

T =

(↑↑ ↑↓↓↑ ↓↓

)

. (5.25)

Multiplying T several times we construct higher spin chains

T 2 =

(↑↑ × ↑↑ + ↑↓ × ↓↑ ↑↑ × ↑↓ + ↑↓ × ↓↓↓↑ × ↑↑ + ↓↓ × ↓↑ ↓↓ × ↓↓ + ↓↑ × ↑↓

)

=

(↑↑↑ + ↑↓↑ ↑↑↓ + ↑↓↓↓↑↑ + ↓↓↑ ↓↓↓ + ↓↑↓

)

. (5.26)

The taking of the trace is the result of the boundary condition s1 = sN+1. It is easy to see thatthis construction leads to the sum of all the configurations for a given N . The eigenvalues ofthe transfer-matrix give us the solution for the partition function Z. In fact

Z (F, h) = Tr(TN

)=

∑

i

λNi . (5.27)

but we are more interested in the result for the free energy

f (F, h) = limN→∞

1

Nln (Z (F, h)) = lim

N→∞1

Nln

[∑

i

λNi

]

= limN→∞

1

Nln

[

λNMax

∑

i

(λi

λMax

)N]

= lnλMax, (5.28)

70

as the quantity(

λi

λMax

)Nvanishes in the thermodynamic limit, except for the value i = Max11.

Thence, the highest eigenvalue gives us the free energy of the system. Let us diagonalize T

|T − λI| =

∣∣∣∣∣

eF+h2 − λ e−F

e−F eF−h2 − λ

∣∣∣∣∣

= λ2 − 2λeF coshh

2− e2F

(e−4F − 1

)

λ =

eF

(

cosh h2 +

√

e−4F + sinh2 h2

)

eF

(

cosh h2 −

√

e−4F + sinh2 h2

) (5.29)

Only the highest eigenvalue contributes to the free energy in the thermodynamic limit. There-fore, the final result is

f (F, h) = F + ln

(

coshh

2+

√

e−4F + sinh2 h

2

)

. (5.30)

By using the free energy, any observable can be easily worked out. For instance, accordingto equation (5.6), the result for the magnetization is

〈m〉 =∂f

∂ h2

=

sinh h2 +

sinh h2

cosh h2

q

e−4F +sinh2 h2

cosh h2 +

√

e−4F + sinh2 h2

=sinh h

2√

e−4F + sinh2 h2

. (5.31)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Mag

net

izat

ion〈m

〉

Magnetic field h

Figure 5.3: Magnetization density as a function of the external field h for the one-dimensionalIsing model. F was set to F = −0.50; as F < 0, we are dealing with an antiferromagneticcoupling.

At vanishing magnetic field and non-zero temperature, 〈m〉 is always zero, so no spontaneoussymmetry breaking happens for the one-dimensional model, as Ising proved in 192512.

11The eigenvalues are bounded for finite F (or, equivalently, non-zero temperature), so λi

λMax

≤ 1. Unboundeigenvalues do not give rise to a well defined free-energy.

12Only at zero temperature (F → ∞, a trivial fixed point) the transition happens.

71

Equation (5.31) for the magnetization is completely general, which means that we can applyit to the case of a pure imaginary magnetic field h = iθ. For θ = π, the Z2 symmetry is restored(the magnetic field effect amounts to a sign σ, depending on the ‘topological charge’ M/2 of theconfiguration σ = eiπM/2; this sign is invariant under a Z2 transformation). Then, the questionis whether there exist spontaneous symmetry breaking or not. The answer was given in (5.31):Substituting h → iθ

〈m〉 =i sin θ

2√

e−4F − sin2 θ2

. (5.32)

0.0

0.1

0.2

0.3

0.4

π4

π2

3π4

π0

Mag

net

izat

ion−

i〈m〉

θ angle

Figure 5.4: Magnetization density as a function of the θ angle for the one-dimensional Isingmodel. F was set to F = −0.50.

And the magnetization takes a non-zero expectation value for the one-dimensional Isingmodel at θ = π, a fact that indicates spontaneous Parity breaking. It is quite remarkable thatthe Ising model within an imaginary external field is not properly defined with ferromangeticcouplings [81]. Setting F > 0 we find that the denominator of 5.32 explodes if

e−4F = sin2 θ

2,

holds, and the free energy

f (F, θ) = F + ln

(

cosθ

2+

√

e−4F − sin2 θ

2

)

. (5.33)

becomes undefined for certain values of θ, for the argument of the logarithm may vanish ifF > 0. Hence, we will deal with the antiferromagnetic Ising model (F < 0) from now on, unlessit is explicitly stated.

72

5.3 Systems with a topological θ term

Ways to simulate complex actions

Why analytic continuations fail in this case

For dimensions higher than one, the model within an external magnetic field is not exactlysolvable13. Simulations can still be performed for real values of the reduced magnetic field, butthe sign problem prevents us to do the same in the case of an imaginary external field. Onewell-known attempt to overcome this obstacle is the use of analytic continuations.

In the absence of phase transitions, the order parameter m (h) should be a smooth functionof the external field h without singularities. If m (h) is measured for enough real values of h ata very high precision, a careful extrapolation to imaginary fields should be possible14, providedan ansatz function for the behaviour of the order parameter. The kind of extrapolation hasbeen a research topic during the last years, mostly in the framework of finite density QCD (see[82, 83, 84, 85, 86], also [87] and references therein).

A great caveat of this method is evident: It does not allow us to explore neither the criticalpoints nor the region lying behind a phase transition. We are doomed to stay in the same phase,for the analitical prolongation does not account for the singularities and discontinuities linkedto a phase transition. For the particular case of θ-vacuum, this fact should not be a majorproblem, as only transitions at the edge (θ = π) are expected to happen15.

Unfortunately, there is another serious problem: as a priori the behaviour of the models inthe θ 6= 0 is unknown, there is not a way to provide a sensible ansatz for the fitting functionof the order parameter. In fact, it can be proved that two functions differing slightly in thereal plane can diverge exponentially in the complex plane, and the results are bound to dependstrongly on the ansatz functions. Therefore, the analytical continuations are not reliable forthis particular problem –unless we only want to know the behaviour of the order parameter inthe neighbourhood of θ = 0–, and we need other sensible methods to compute the θ dependenceof the theories.

The p.d.f. computation at θ = 0

On the other hand, the problem of θ-vacuum is set up in a way that suggests quite strongly aparticular kind of dependence in θ. The partition function of any system in the presence of a θterm is periodic, and can be decomposed in sectors of different topological charge n as

ZV (θ) = ZV (0)∑

n

pV (n) eiθn, (5.34)

which resembles the Fourier transform of the probability distribution function (p.d.f.) of thetopological charge at θ = 0. The probability of the topological sector n is therefore given bypV (n), and this quantity can be measured from simulations at imaginary values of θ (i.e. realvalues of the external field h = −iθ). Unfortunately, this is a very difficult task, for

i. Precision in a numerical simulation can not be infinite, is limited by statistical fluctuations.Thus the measurement of pV (n) suffers from errors.

ii. Small errors in pV (n) induce huge errors in the determination of ZV (θ), for this quantityis exponentially small ZV ≈ e−V due to the sign problem.

13Onsager’s solution is only valid for h = 0.14Complex analysis proves that the analytic continuation must exist and be unique.15Nonetheless, we will see in the Ising model a notable exception to this rule.

73

iii. Even if we were able to evaluate pV (n) with infinite accuracy, the terms on the sum (5.34)differs by many orders of magnitude (from 1 to e−V ).

In fact, the different groups that have tried to determine with high precision the p.d.f. of thetopological charge by standard simulations, or by more sophisticated methods (re-weighting ormultibinning techniques), have found artificial phase transitions16 in the U(1) model and in theCPN models. The one to blame of these ghost transitions is the flattening of the free energyfor θ-values larger than certain threshold. In [88, 89], this threshold is roughly evaluated, andthe flattening behaviour demystified:

At zero external field and finite volume, usually the vanishing topological sector dominates.This implies that during the measurements

|δpV (0)| ≥ |δpV (1)| ≥ |δpV (2)| . (5.35)

Of course, this is not a theorem; in fact, the error in the determination of higher sectors mightcompete with that of sector zero. This statement only reflects what usually happens duringsimulations.

Thence, the most prominent error in the computation of the free energy is δpV (0), and theθb threshold is given by

f (θb) ≈1

V|ln |δpV (0)|| . (5.36)

Consequently, if δpV (0) > 0, then the free energy is given by a constant value

f (θ) ≈ 1

Vln δpV (0) θ > θb. (5.37)

A wavy behaviour can appear if the errors in other modes (n = 1, 2 . . .) are competing with thezero mode δpV (0). If, on the other hand, δpV (0) < 0, the free energy becomes impossible tobe measured for θ > θb.

This way, the precision in the measurements limits how far can we go with θ. These areno news to us, but it is remarkable the severity of the problem: A reliable computation of theorder parameter (or the free energy) to all values of θ is not feasible by direct measurement ofthe p.d.f., due to the huge statistics required.

That is why other approaches (or at least, serious refinements of the p.d.f. approach) shouldbe considered. The method I describe in the following lines succeeded to reproduce the θdependence of the order parameter for those systems where m (θ) was a non-decreasing functionin the 0 ≤ θ ≤ π range.

An improvement over the standard p.d.f. methods

Let’s write the partition function of any system with a topological charge term as a sum (integralin the infinite volume limit) over the density of topological charge xn = n/V ,

ZV (θ) = ZV (0)∑

n

e−V fV (xn)eiθV xn , (5.38)

where we have set pV (n) = e−V fV (xn). Assuming that CP -symmetry is preserved at vanish-ing17 θ, we expect e−V fV (xn) to approach a delta distribution centered at the origin in thethermodynamic limit.

16Induced by the simulation method, or by the way the order parameter is computed, therefore this is not areal phase transition. Lack of precision, bad p.d.f. determinations due to poor sampling or rounding errors aresome of the problems leading to these ghost transitions.

17Otherwise, the theory would be ill-defined at θ 6= 0.

74

In the thermodynamic limit, fV (xn) becomes f (x). As all the coefficients entering inequation (5.38) are positive, the free energy for imaginary θ (that is, real field h = −iθ) isgiven in the thermodynamic limit by the saddle point. We assume in the following that the firstderivative of f (x) is well-defined, except at most in isolated points. Then, we can write

f ′ (x) = h, (5.39)

so the external field is given as a function of the topological charge density. The procedure toreconstruct f (x) from (5.39) is the following

i. The mean value of the topological charge as a function of an external real field is measuredwith high accuracy (within a fraction of percent). The saddle point equation (5.39) isapplied afterwards to obtain f ′ (x).

ii. The function f ′ (x) is fitted to the ratio of two polynomials. Then, the integral to computef (x) is performed analytically. This step is essentially different to what other groupsproposed, and it solves the problems of the θ threshold in some systems.

iii. Finally, a multi-precision algorithm is used to calculate the partition function directly from(5.38), using the function f(x) computed in point (ii).

Although f (x) suffers from two different sources of errors, statistical and systematic comingfrom finite volume corrections to the saddle point equation (5.39), only statistical errors wereimportant18 in the determination of f (x) in the whole range of θ’s. To see why, we must analyzecarefully the behaviour of fV (xn) under correlated and uncorrelated errors.

If we call fV (xn) the exact value of this function at a given volume V , and ∆fV (xn) theerror, then the partition function computed with errors Z ′

V (θ) is related to the exact partitionfunction ZV (θ) by

Z ′V (θ) = ZV (θ)

⟨

e−V ∆fV (xn)⟩

. (5.40)

Suppose that the error function ∆fV (xn) is an extremely uncorrelated function, vanishingeverywhere except in a given value xm. In this case, the error in the computation in thepartition function is given by

⟨

e−V ∆fV (xn)⟩

= 1 + 2e−V (fV (mV )−gV (θ))

(

e−V ∆fV (mV ) − 1

)

cos (mθ) , (5.41)

where gV (θ) is assumed to be the exact free energy density. As gV (θ) increases with θ, the firstexponential on the r.h.s. of (5.41) is doomed to be small near the origin, spanning correctionsof order 1

V . Unfortunately this pictures changes as θ increases, and the partition function mightbecome negative.

On the other hand, if the errors are largely correlated, assuming for instance that ∆fV (xm) isalmost constant, the errors propagated to gV (θ) are almost constant as well, by virtue of (5.41);these errors amount to an irrelevant constant in the free energy. This picture is encouraging, forthe fitting of (5.39) to a ratio of polynomials to perform the integration analitically is expectedto produce largely correlated errors. Indeed, these ideas were successfully tested [90] in theone-dimensional Ising model and the two-dimensional U(1) model, and it was used to predictthe behaviour of the CP 3 model. The results were impressive by that time, solving completelythe problem of the flattening of the order parameter beyond the critical value of θ. The keyof this success was the aforementioned correlation among the errors: Test performed using the

18This statement was valid for the models studied in [90].

75

same method an adding an apparently negligible 0.1% random error to the measured free energyf (x) led to disaster. Nonetheless, if the error was correlated, it could be as large as the 50% ofthe original value, and the final result would be quite reasonable.

Figure 5.5: Effects of correlated and random errors on the θ dependence of the topologicalcharge Q in the two-dimensional U(1) model at β = 0. The continuous curve is the exact result,the dashed curve is the result obtained by substituting f(x) by f(x)(1 + 1

2 sin(x2)), and thedotted curve is the result obtained by adding a random error of the order of 0.1% to f(x).

However, this is not the final method to find out the θ dependence of the different models.The flattening behaviour can appear -and in fact, it does- whenever the behaviour of the orderparameter is not monotonous. This is not a prediction, but seems to be a general rule. Theflattening was first observed in a simple toy model which featured symmetry restoration atθ = π

f (θ) = ln (1 + A cos θ) . (5.42)

For this model, the order parameter vanishes only at θ = 0, π

−im (θ) =A sin θ

1 + A cos θ, (5.43)

but the method predicted an almost flat behaviour beyond the point θ = π2 .

The problem might be solved by using a more suitable fitting funtion. In fact, we were ableto reproduce qualitatively the behaviour of the order parameter in the two-dimensional Isingmodel, which –anticipating the result obtained in the following pages– features a vanishing orderparameter at θ = π.

The problem is the fact that the proper fitting function is not always integrable analitically.An odd polynomial always reproduces correctly the behaviour of the function m (h), but we areseeking the solution to the saddle point equation (5.39), h (m), which is the inverse. The exactsolution to this inversion can be expressed in a closed form for polynomials of order 3 or less,

76

0

0.001

0.002

0.003

0.004

0.005

π4

π2

3π4

π0

Ord

erpar

amet

er−

i〈m〉

θ

Exact resultP.d.f.

+

+

+

++

++

++

++++ + + + + ++

Figure 5.6: Attempt to solve the toy model defined in (5.42) using the improved p.d.f. method.The feared flattening appears at θ ∼ π

2 .

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1.4e-05

1.6e-05

0π4

π2

3π4

π

Ord

erpar

amet

er−

i〈m〉

θ

P.d.f.

++

++

++

++

++

++++

++++

++++++++++++++++++++

++

++

++

++

++

+

+Expected result

××××××××××××××××××××××××××××××××××

×××××××××××

××

×

×××××××××××

××

×

×××××××××××××××××××××××××××××××××××××××××××××××××× ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××

××××××× ××××××××××××××

×××××××××××××××××××××××××××××××××××××××××××××××

×××××××××××××××××

×××××××××××××××××××××××××××××××××××××××××××××××

×××××××××××××××××

××××××××××××××××××××××××××××××××××××××××××××××××

××××××××××××××××

××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

×

Figure 5.7: Ising’s vanishing of the order parameter using the improved p.d.f. method (crosses).The numerical results differ notoriously to those obtained in the next sections (circles), nonethe-less the qualitatively correct behaviour of the order parameter suggest that further refinementsmight work properly. Errors were not estimated for the p.d.f. method.

but the resulting function is not analitically integrable. A numerical approach to the problemintroduces uncorrelated errors which spoil all the good properties of this approach. In the lastexample of fig. 5.7, an approximation to the most suitable fitting function was used, but clearlyit was not enough to predict the values of m (h) with high precision. In fact, fig. 5.7 representsa very well behaved case, usually the order parameter might differ from the expected case byan order of magnitude, or even give a wrong behaviour. As explained before, the result dependscritically on the chosen fitting function.

This is not the only problem this method has. Another dark point in the method is the useof the saddle point relation, which has corrections at finite volume of order O

(1V

). These are

uncontrolled sources of systematic errors. Even if these errors did not seems to affect too much

77

the final results in the models tested successfully, an upper bound for the errors could not beestimated in any way.

On the whole, although the method proposed in [90] represented a large improvement overwhat existed at that point, it was clear that another approach, one which was capable ofestimating the magnitude of the errors, was necessary. The final conclusion was, further researchshould be made, and that is how the method described in the following sections was created.

Ising’s miracle

As I explained before, the Ising model is a good toy-model for testing algorithms in developmentto simulate θ-vacuum systems in the computer. What was completely unexpected was the factthat the most successful method to deal with a θ term would come from a special property ofthe Ising model. This property was so useful and inspiring that it was called the Ising’s miracle.

To find what is so miraculous about the Ising model, we have to pay atention to equations(5.31) and (5.32). If, instead of 〈m〉, the quotient 〈m〉

tanh h2

is computed, the outcoming function

y (z) =〈m〉

tanh h2

=

(e−4F − 1

)− 12 cosh h

2√

(e−4F − 1)−1 cosh2 h2 + 1

=e−

λ2 z√

e−λz2 + 1. (5.44)

depends only on F and on the variable z = cosh h2 through a special combination of both

e−λ2 z, with λ = − ln

(e−4F − 1

). Due to this somewhat simplified dependency on F and h, the

transformation

yλ (z) = y(

eλ2 z

)

(5.45)

is equivalent to a change in the reduced coupling F , or in the temperature of the model. Theinteresting point here is the fact that this transformation can take the variable z = cosh h

2 to the

range 0 < z < 1 for a negative value of λ, where z becomes a simple cosine z = cos θ2 , implying

that we can measure y (z) for imaginary values of the magnetic field by mean of numericalsimulations at real values of h, which are free from the sign problem.

In other words, for the special case of the one-dimensional model, there exist an infinitenumber of triplets (m, h, F ) such that verify

〈m〉 (F, h)

tanh h2

=〈m′〉 (F ′, h′)

tanh h′

2

, (5.46)

and we can extend this relationship to imaginary values of the magnetic field. The problemof performing a simulation with a complex action is thus reduced to the simulation of a realaction.

In order to check if this property still holds for other systems (for instance, the Ising modelin higher dimensions), simulations at different values of the external field have to be performed.Assuming y (z, F ) = y (g (F ) z), then

∂y

∂F=

∂y

∂ (g (F ) z)g′ (F ) z, (5.47)

∂y

∂z=

∂y

∂ (g (F ) z)g (F ) , (5.48)

∂y∂F∂y∂z

=

∂y∂(g(F )z)g

′ (F ) z

∂y∂(g(F )z)g (F )

=g′ (F ) z

g (F ). (5.49)

78

For the miracle to be preserved in the model, the ratio ∂y∂F /z ∂y

∂z should be independent ofh. Let’s see how we can work this quantity out in a computer simulation. First we need tocalculate ∂y

∂F and ∂y∂z in terms of quantities we know:

∂y

∂F=

∂ mtanh h

2

∂F=

1

tanh h2

∂m

∂F=

1

tanh h2

[⟨(N∑

i

sisi+1

) (N∑

i

si

)⟩

−⟨

N∑

i

sisi+1

⟩ ⟨N∑

i

si

⟩]

, (5.50)

z∂y

∂z= z

∂ 〈m〉tanh h

2

∂ h2

∂ h2

∂z= χ − 〈m〉

tanh h2 sinh2 h

2

, (5.51)

For the one-dimensional Ising model, the simulations reveal the constant ratio over a largerange of fields h (see fig. 5.8).

1.8

1.9

2.0

2.1

2.2

2.3

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

−g′ (

F)

g(F

)

External Field h

♦ ♦ ♦ ♦ ♦ ♦ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

+ + + + +++++++++++++++++++++++++

¤ ¤ ¤ ¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

× × × × ×××××××××××××××××××××××××△ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△

Figure 5.8: Ising’s miracle check along formula (5.51). The continuous lines represent theanalytical result, while the crosses stand for the numerical data. We performed short simulations(only ∼ 100000 iterations) for several values of F in a L = 100 lattice. Errors are smaller thansymbols.

Unfortunately, this property is exclusive of the one-dimensional case. For two dimensions,the order parameter 〈m〉 can not be written as a function of g (F ) z and the ratio shows a slightlydependence on the reduced magnetic field. For three dimensions, the dependence becomes a bitmore pronounced. The peak in 5.9 is produced by the antiferromagnetic-ferromagnetic phasetransition19.

Computing the order parameter under an imaginary magnetic field

Although Ising’s miracle is absent in higher dimensions, we can still take advantage from themethodology it profiles. For the one-dimensional case, a measurement of the order parameter

19The antiferromagnetic Ising model displays, for strong enough couplings, a phase transition at non-zeroexternal magnetic field: the spin-coupling tries to put the system in an antiferromagnetic state, whereas theexternal field tries to order the spins in a ferromagnetic fashion. As the value of the external field increases, theferromagnetic behaviour takes over.

79

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

External Field h

Ising 2D F = -0.25

♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦♦♦♦♦♦♦♦♦♦♦♦♦

♦Ising 2D F = -0.75

+ + + + + + + + + + ++++++++++++++++

+Ising 3D F = -0.15

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ ¤¤¤

¤Ising 3D F = -0.30

× × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

×

Figure 5.9: Ising’s miracle realization in 2D and 3D. The miracle is approximate in 2D and 3Dfor low values of the field. The statistics of the simulations are 100000 iterations, and the latticelengths are, for 2D L = 50 and for 3D L = 25. Errors are smaller than symbols.

produced at the point (F, z) is equivalent to a measurement done at (F ′, z′) if the followingrelationship holds

g (F ) z = g(F ′) z′,

g (F ) =(e−4F − 1

) 12 . (5.52)

This way, and choosing carefully the value of F , a simulation performed at a real value ofthe reduced magnetic field z ≥ 1 is equivalent to another simulation performed at imaginaryvalues of h (where z < 1).

The procedure to find out the order parameter at imaginary values of the reduced magneticfield in non-miraculous systems relies on scaling transformations. We define the function yλ (z)as

yλ (z) = y(

eλ2 z

)

. (5.53)

For negative values of λ, the function yλ (z) allows us to calculate the order parameter(tanh h

2 y (z))

below the threshold z = 1.

If y (z) is non-vanishing for any z > 020, then we can plot yλ/y against y up to very smallvalues of y. Furthermore, in the case that yλ/y is a smooth function of y close to the origin,then we can rely in a simple extrapolation to y = 0. Of course, a smooth behaviour of yλ/ycan not be taken for granted; however no violations of this rule have been found in the exactlysolvable models.

The behaviour of the model at θ = π can be ascertained from this extrapolation. The criticalexponent

γλ =2

λln

(yλ

y

)

(5.54)

20Even though the possibility of a vanishing y (z) for some value z > 0 can not be excluded completely, it doesnot happen for any of the analitically solvable models we know.

80

in the limit y (z) → 0 (z → 0) gives us the dominant power of y (z) for values of z close tozero. As z → 0, the order parameter tan θ

2 y(cos θ

2

)behaves as (π − θ)γλ−1. Then, a value of

γλ = 1 implies spontaneous symmetry breaking at θ = π, for at this point the broken symmetryby the external field is restored. A value between 1 < γλ < 2 signals a second order phasetransition, and the corresponding susceptibility diverges. Finally, if γλ = 2, the symmetry isrealized (at least for the selected order parameter), there is no phase transition and the freeenergy is analytic at θ = π21.

We can take the information contained in the quotient yλ

y (y) to the limit, and calculatethe order parameter for any value of the imaginary reduced magnetic field h = −iθ through aniterative procedure.

The outline of the procedure is the following:

i. Beginning from a point y (zi) = yi, we find the value yi+1 such that yλ = yi. By definition,

yi+1 = y(

e−λ2 zi

)

.

ii. Replace yi by yi+1, to obtain yi+2 = y(e−λzi

).

The procedure is repeated until enough values of y are know for z < 1. This method can beused for any model, as long as our assumptions of smoothness and absence of singular behaviourare verified during the numerical computations.

Figure 5.10: Iterative method used to compute the different values of y(z). yλ is ploted as afunction of y using a dashed line in the region where direct measurements are available, and acontinuous line in the extrapolated region. The straight continuous line represents yλ = y.

21Other possibilities are allowed, for instance, any γλ > 1, γλ ∈ N leads to symmetry realization for the orderparameter at θ = π and to an analytic free energy. If γλ lies between two natural numbers, p < γλ < q, p, q ∈ N,then a transition of order q takes place.

81

Numerical work

The first thing to do is to check the correct implementation of the method, for its goodness hasbeen already tested in a variety of models [91, 92, 93], and its validity, whenever the assumptionsrequired are fullfiled, is beyond any question. The best way to verify the implementation is toapply the method to a solvable toy model, and compare the results with analytical formulae.In our case, we tested the method in the one-dimensional Ising model, and checked the resultsagainst (5.32).

The simulations were performed at a fixed volume, N = 1000 spins, and fixed reducedcoupling F = −2.0. As in the one-dimensional Ising model there are no phase transitions,and the Ising’s miracle is realized, there is no point in checking the method for several valuesof the reduced coupling. The parameter we varied was the reduced magnetic field h. Asthe simulations were done quite fast, we could obtain data for many values of h with largestatistics. In fact, for each point in the plots we performed 107 metropolis iterations. In orderto reduce autocorrelations, we performed at each iteration two sweeps over the lattice, proposingmetropolis changes in the spins. The plots for the critical exponent and the order parameterare shown in fig. 5.11 and 5.12.

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.00 0.05 0.10 0.15 0.20 0.25

2 λln

yλ y

y

1D Ising Model F = -2.0

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + + + + + +

+Quadratic Fit

Figure 5.11: Calculation of the critical exponent γλ. The crosses correspond to the numericalsimulation data, whereas the line is a quadratic fit. The extrapolation to zero seems quitereliable, as the function is smooth enough. Errors are smaller than symbols.

Our result for the critical exponent from the fit in fig. 5.11 is

γλ = 0.99980 ± 0.00008,

which agrees with the analytical result. The fact that we obtained the right results in the one-dimensional model is not a big deal, as this model is miraculous, then it is expected to behavewell under the scale transformations defined in this method.

Then we simulated higher dimensional models, expecting to see departures from this be-haviour, as these models feature phase transitions between ordered (antiferromagnetic) anddisordered phases. The two-dimensional simulations were done in a 1002 lattice, after 100000termalization sweeps. We spent 5000000 steps to measure each point accurately. The three-dimensional case, on the other hand, used a 503 volume, and measured each point for 2500000steps after 100000 steps of thermalization. The outcoming results showed the expected depar-

82

0.000

0.005

0.010

0.015

0.020

0.025

0π4

π2

3π4

π

Ord

erpar

amet

er−

i〈m〉

θ

1D Ising model F = -2.0

++

+

++

++++++

++

++

+++++++++++++++++

++

++

+

++++++++++++++++++++++++++

++

+

+

++ ++++++++++++++++++++++++++++++++

+

++ +++++++++++++++++++++++++++++++++

++

+

+

+++++++++++++++++++++++++++++++++++++++

++

++

+

++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+Analytic result

Figure 5.12: Order parameter as a function of θ. The non-zero value of the order parametermarks the spontaneous breaking of the Z2 symmetry at θ = π.

ture in the behaviour. Our prediction for the critical exponent γλ ≈ 2 reveals a vanishing orderparameter at θ = π in the ordered phase (F = −1.50 for 2D and F = −1.00 for 3D)22.

γ2Dλ = 1.9986 ± 0.0014

γ3Dλ = 1.9998 ± 0.0002

We can confirm this facts by plotting the order parameter against θ, as it is done in fig. 5.16and 5.17.

1.60

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2.10

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

2 λln

yλ y

y

2D Ising Model F = -1.5

+++

+

+++ ++ + + +

++++++++

++++++++++++++++++++++

+

+++++++++++++++++++ ++ + +

+ +

+++++++++++++++++++++++++++++++++++++++++++++++++++ + + +

+

+++++++++++++++++++++++++++++++++++++++++++++++++++ ++ + + + + + +

+

+++++++++++++++++++++++++++++++++++++++++++++++++ + + + + + +

+++++++++++++++++++++++++++++++++++++ ++++++++++++

+Cuadratic Fit

Figure 5.13: Calculation of the critical exponent γλ in the ordered phase of the two-dimensionalmodel. The pluses correspond to the numerical simulation data, whereas the line is a cuadraticfit. Errors are much smaller than symbols.

22Actually the Z2 symmetry is spontaneously broken, for the staggered magnetization mS 6= 0 [94]. This pointwill be clarified in the mean-field approximation.

83

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

0 5e-05 0.0001 0.00015 0.0002 0.00025

2 λln

yλ y

y

3D Ising Model F = -1.0

+++++++++ ++++++++ +++++++++ +++++++++++++++ +++++++++++++++++++ +++ +

+Constant Fit

Figure 5.14: Calculation of the critical exponent γλ in the ordered phase of the three-dimensionalmodel. The pluses correspond to the numerical simulation data, whereas the line is a constantfit. Errors are much smaller than symbols, except for the points lying close to the origin.

0.5

1.0

1.5

2.0

2.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2 λln

yλ y

y

2D Ising Model F = -1.5

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++ + + + ++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+++++ + ++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

++

+++++ ++ + +++++++++++++++++++++++++++++++++++++++++++++++++++

+

Figure 5.15: Due to the peaked behaviour of the quotient yλ/y around y = 0.3, the extrapolationto zero required many simulations at small values of the magnetic field. Errors are much smallerthan symbols.

The disordered phase revealed a caveat of this method, as it was impossible for us to extrap-olate the function yλ

y (y) to zero. The reason is simple: at small values of F , y and yλ approach

the unity, for at vanishing F we recover the paramagnetic Langevin solution m = tan θ2 , which

will be derived later. The smaller the value the F , the greater the gap between zero and ourdata becomes, and at some point, the extrapolation is not reliable any more, and the outcomingresults depend strongly on the fitting function used. An example can be seen in fig. 5.18, wherethe data for the two-dimensional model at F = −0.40 are plotted. In this case, we are to farfrom zero to find out accurately the critical exponent, and the value of F could not be loweredmuch more, for the transition to the ordered phase is known to happen at F ∼ −0.44. In 5.19a similar example is shown for the ordered phase in the three-dimensional model, but this time

84

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1.4e-05

1.6e-05

0π4

π2

3π4

π

Ord

erpar

amet

er−

i〈m〉

θ

2D Ising model F = -1.5

++++++++++++++++++++++++++++++++++

+++++++++

++

+

+

+

+++++++++

+++

++

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++

++++ ++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

++++

++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++

+++

++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

Figure 5.16: Order parameter as a function of θ in the ordered phase of the two-dimensionalmodel. It vanishes at θ = π.

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1.4e-05

0π4

π2

3π4

π

Ord

erpar

amet

er−

i〈m〉

θ

3D Ising model F = -1.0

++

+

+

+

+

++++++++

++++++++++++++++

+++++

+++++

++++++++

+++++++

++

++++++++

+++++++

++

++++++++

+++++++

++

++++++++

+++++++++

+++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

Figure 5.17: Order parameter as a function of θ in the ordered phase of the three-dimensionalmodel. As in its two-dimensional counterpart, it vanishes at θ = π. Errors are smaller thansymbols.

a tentative extrapolation could be done, casting a reliable result.

This examples show how this method works fine when the antiferromagnetic couplings arestrong enough. In general, the method performs well for asymptotically free theories, whosecontinuum limit lie in the region of weak coupling. In this region, the density of topologicalstructures is strongly suppressed. Thus, the order parameter, and hence, y (z), take small values,making the plot yλ

y (y) easily extrapolable to zero. In the particular case of the antiferromagneticIsing model, the behaviour seems to be the contrary, for the coupling F < 0 opposes to theformation of topological structures, and large values of |F | ensure a small magnetization. Ahigh value of the dimension also helps, for instance, the three-dimensional model requires alower value of the coupling than the two-dimensional case to make a reliable extrapolation ofyλ

y (y) to y → 0, for each spin is affected by a higher number of neighbours, so the topological

85

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2 λln

yλ y

y

2D Ising Model F = -0.4

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

Figure 5.18: Failed calculation of the critical exponent γλ in the disordered phase for the twodimensional model. Our data is so far from the y = 0 axis, that an extrapolation can not beused to find out the value of γλ. A peak for lower values of y, as the one appearing in fig. 5.15,cannot be discarded ‘a priori’. Errors are much smaller than symbols.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 λln

yλ y

y

3D Ising Model F = -0.3

+++ +++++ ++++

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

++

+

++

+

+

+++++++++++++++++++ +++++++++++++++++++++++++++++++++++

+

+

+

+++++++++++++++++

+++++++++++++++++++++++++++++++

+

+++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+Quadratic Fit

Figure 5.19: Another calculation of the critical exponent γλ in the ordered phase F = −0.3for the three-dimensional model. Our data approaches the y = 0 axis enough to try an ex-trapolation, but the result suffers from much larger errors than in the F = −1.0 case. Hereγλ = 2.079 ± 0.003, and the measurement errors are much smaller than symbols.

structures are even more supressed.

As this method failed to deliver interesting results in the disordered phase, we tried a differentapproach: we expected naively that the two-dimensional model resemble the one-dimensionalmodel at low values of the coupling. Since the p.d.f. method described in this chapter workedwell for the one-dimensional case [90], it made sense that we applied it to the present scenario.What we found is an unstable behaviour: sometimes the method seems to predict the phasetransition, in the sense that at finite volume there is not true phase transition, and an abruptmodification in the order parameter, linking the two expected behaviours, should happen. This

86

is what we observe in one of the data sets of fig. 5.20. Nonetheless, if a slightly different set ofpoints is taken to fit the saddle point equation (5.39), the resulting data show a sharp departurefrom the expected behaviour at some θ.

0.00

0.05

0.10

0.15

0.20

0.25

0π4

π2

3π4

π

Ord

erpar

amet

er−

i〈m〉

θ

Original fit

++++++++++++

+++++++++++++++++++++++++++++++++++++

+Slight modification

¤

¤

¤

¤

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

¤

Figure 5.20: Failed calculation of the order parameter in the disordered phase using the p.d.f.method. In the first case, the points seem to predict a phase transition at θ ∼ 2.35, whereas inthe second case the points depart sharply from a smooth function and never come back. Theonly difference between fits was the number of points used: in the second case, only half ofthe points (the closest to the origin) were used. Other modifications in the fitting proceduresindicate us that the transition point is not stable. This might indicate either a failure in thefitting function, or a phase transition, and the impossibility for the method to precise thetransition point, unless a perfect ansatz is made. Errors were not estimated.

There are two possible explanations to this behaviour: either the fitting function selected iscompletely wrong, or there is some hidden phenomena we are overlooking. The fitting functionused was an odd quotient of polynomials

ax3 − x

cx2 − b

which should account perfectly for the behaviour of the order parameter, given the assumptionthat it is similar to the one-dimensional case. The addition of more terms to the fit did not domuch to improve the result, hence this possibility was discarded.

The existence of a phase transition in the middle, however, was an interesting option. Indeed,the two-dimensional model in the presence of a θ term was solved exactly at the point θ = πalmost sixty years ago by Yang and Lee in [94], and reviewed again in [95]. In those papers,a phase diagram was proposed were the antiferromagnetic model always stayed in an orderedphase at any non-zero value of F . Since the system is in a disordered state for low F ’s andzero field, some phase transition has to occur in the middle. Thence, the failure of the p.d.f.method should be due to a poor ansatz for the fitting function, caused by the presence of aphase transition at some θc.

The fact that the results for the two- and three-dimensional models are qualitatively thesame in the ordered phase, makes us wonder whether for some value of the dimension D > 3this behaviour is not observed. Moreover, the behaviour of this model in the disordered phaseis completely unknown to us. That is why we decided to carry out a mean-field approximation

87

of the model, and compute the critical exponent γλ. As we know, mean-field results for othercritical exponents are exact for the n-dimensional ferromagnetic Ising model, provided thatn ≥ 4. Thus we expect that, if the mean-field result for γλ is the same to that of the two- andthe three-dimensional Ising model, then γλ = 2 for any value of the dimension.

5.4 The Mean-field approximation

An introduction to Mean-field theory

The mean-field approximation was introduced by Weiss in 1907, just two years after the ap-pearance of Langevin’s theory of paramagnets. Langevin’s theory explains the paramagneticphenomena assuming that paramagnetic materials are composed of a set of disordered magneticdipoles23. The interaction among these dipoles is weak enough to be negligible against thermalfluctuations, and the only ordering agent is the external magnetic field,

H (B, {si}) = −BN∑

i

si. (5.55)

The model is easily solvable: As the spin do not interact with each other, the partitionfunction factorizes:

Z (h) =∑

{si}e−βH(h) =

[∑

si=±1

eh2si

]N

= 2N coshN h

2. (5.56)

The free energy is given by

f (h) = ln 2 + ln coshh

2, (5.57)

and the magnetization can be written in a closed form

m (h) = tanhh

2. (5.58)

The theory was quite a success for paramagnets, as it was able to predict Curie’s law inits naive form χ = C/T . Nevertheless, it could not explain ferromagnets, and spontaneousmagnetization phenomena. The reason is quite straighforward: As the spin system describe byLangevin’s theory is non-interacting, at zero external field, the magnetization should vanish.This can be readily checked in (5.58).

Weiss hypothesis improved Langevin’s theory assuming a crude interaction among the spins.Each spin created a small magnetic field, called molecular field, which reinforced the effect of theexternal field. Once the external field is removed, the molecular field might be strong enoughto give rise to spontaneous magnetization phenomena. The way to treat the molecular field wasexcessively simple: The spins interacted with an average magnetic field created by all the otherspins. This effect is included on (5.55) just by writting

H (J, B, {si}) = −JN∑

{i,j}si〈sj〉 − B

N∑

i

si, (5.59)

23Originally, Langevin’s theory was applied to classical vectors, instead of classical spins S = 1/2. Here we willapply the same ideas in the framework of the Ising model.

88

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 0.5 1.0 1.5 2.0

Mag

net

izat

ion〈m

〉

Magnetic field h

Figure 5.21: Langevin’s prediction for the magnetization density distribution function of anS = 1/2 paramagnet.

which resembles the Hamiltonian for the Ising model (5.1). By using invariance under transla-tions and the definition of the magnetization m = 1

N

∑Ni si we find

H (J, B, m) = −NJm2 − NBm. (5.60)

So the mean-field hamiltonian is written as

H (J, B, m) = −N(Jm2 + Bm

). (5.61)

We could have derived (5.61) directly from the infinite coupled Ising model, described by

H (J, B, {si}) = − J

N

N∑

i,j

sisj − BN∑

i

si. (5.62)

Two comments are in order:

• The mean-field hamiltonian depends only on the magnetization m.

• A constant coupling J in (5.62) would lead to a N2 divergence in the thermodynamiclimit, and the energy would not be an extensive quantity. The reason is quite clear: Eachspin is coupled to all the other spins24 with the same strength. In the thermodynamiclimit, this kind of coupling diverges, unless the coupling constant becomes increasinglysmall

J → J

N.

As we can see in (5.62), the dimensionality of the system is completely irrelevant here,as every spin couples with every other spin. In fact, this system seems to feature an infinitedimension, for the coordination number25 of each spin is, in an effective way, infinite in the

24Being rigorous, each spin is coupled to the other N − 1 spins, but the difference is negligible in the thermo-dynamic limit.

25The coordination number q is the number of neighbours. In systems with nearest-neighbour interactions, itis an interesting property, and for the case of an square lattice, it is proportional to the dimensionality of thesystem q = 2D.

89

thermodynamic limit. This way to understand the mean-field approximation gives us someinsight on what to expect from it. We can guess now that this approximation works better inhigher dimensions, and in models with a great number of interacting neighbours per site. In anycase, the approximation is very crude, for it does not depend on the dimension of the systemwe are considering26.

Mean-field theory for the antiferromangetic Ising model

The Hamiltonian (5.59) refers to the ferromagnetic case. We could naively think that fornegative values of the coupling J , this Hamiltonian is applicable to the antiferromangetic case,but this is not true. The problem lies in the fact that every spin couples to every other spin inthe same way. An antiferromagnetic coupling in (5.59) would lead to a system where each spintries to align in an opposite way to all the other spins, introducing frustration in the model.However, in antiferromagnetic compounds the spin-alignment pattern is staggered, like a chessboard, and frustration is absent. The short-ranged model also lacks frustration for an evennumber of spins27. This is an essential difference, which may lead to qualitatively differentbehaviours. Thence, in order to define properly the mean-field version of the antiferromagneticIsing model, we should divide the lattice in two sublattices, and define a coupling among spinswhose sign depends on whether these two spins are on the same sublattice or not.

Figure 5.22: Representation of the two sublattices in a two-dimensional lattice. The lattice S1

is indicated by the circles, whereas the lattice S2 is marked by crosses. Note that, if there isonly a nearest neighbour interaction, the circles do not interact among each other, but onlywith the four crosses that surrounds each circle. The antiferromagnetic mean-field approachmust distinguish whether a circle/cross is interacting with a cross or a circle, and vice-versa.

For spins belonging to the same lattice, the coupling should be ferromagnetic (J > 0),but for spins belonging to different sublattices, the coupling should favour antiparallel ordering(−J < 0), according to the antiferromagnetic nature of the system. Therefore, two differentmean-fields should appear, 〈si〉i∈S1

= m1 and 〈si〉i∈S2= m2, referring to each one of the different

sublattices. The corresponding Hamiltonian is a bit different than (5.62)

H (J, B, {si}) = −N[

J (m1 − m2)2 − B (m1 + m2)

]

=

= − J

N

∑

i∈S1

si −∑

j∈S2

sj

2

− B1

∑

i∈S1

si − B2

∑

j∈S2

sj , (5.63)

where we have separated the external magnetic fields acting on the different sublattice forcomputation purposes. In the end, the equality B1 = B2 is set. In this system, ferromagnetic

26In fact, a modification in the coordination number (which is the only dependence on the dimension the systemmight display) in a mean-field approximation for a given model only amounts to a rescaling of the coupling J . Ifwe use the hamiltonian (5.62), the concept of coordination number does not make much sense anyway.

27With periodic boundary conditions.

90

interactions are stablished among the spins of the same sublattice, whereas antiferromagneticforces appear among spins of different sublattices.

Let us solve this model. The partition function

Z (F, h) =∑

{si}e

FN

“

P

i∈S1si−

P

j∈S2sj

”2

+h12

P

i∈S1si+

h22

P

j∈S2sj (5.64)

can be summed up by applying the Hubbard-Stratonovich identity28 to linearize the quadraticexponent

Z (F, h) =1

π12

∫ ∞

−∞

∑

{si}e−x2+2x

"

|F |12

N12

+h1

#

P

i∈S1si−

"

|F |12

N12

−h2

#

P

j∈S2sj

dx. (5.65)

At this point we see that the introduction of the θ term through the transformations

h1 → iθ1, h2 → iθ2,

render the hyperbolic cosines complex. The 12 factor is allows us to define properly the quantized

number M2 . The integrand factorizes, as there is no spin-spin interaction

Z (F, h) =2NeF

π12

∫ ∞

−∞e−x2

[

cosh

(

2x|F |

12

N12

+ iθ1

2

)

×

cosh

(

2x|F |

12

N12

− iθ2

2

)]N2

dx. (5.66)

Now we bring the transformation

x → N12 y

dx → N12 dy

so (5.65) becomes

Z (F, h) =2NN

12

eF π12

∫ ∞

−∞

[

e−y2+

12

lnh

cosh“

2|F |12 y+i

θ12

”

cosh“

2|F |12 y−i

θ22

”i]N

dy. (5.67)

where we have written the whole integral as an exponential. We can not solve the integral(5.67), but by using the saddle-point technique29 we should be able to evaluate the free energy.The problem here lies in the validity of the saddle-point approach: were the argument of thelogarithm in (5.67) real, then the saddle point technique would be completely valid, but theargument is, in general, complex30. Fortunately, the argument of the logarithm in (5.67) is realfor the case θ1 = θ2 = θ, as can be demonstrated by using some trigonometric transformations:

cosh

(

2 |F |12 y + i

θ

2

)

cosh

(

2 |F |12 y − i

θ

2

)

=

28See (B.4) Appendix B29See Appendix B.30There exists a complex version of the saddle-point technique, but it has several limitations, and seldom does

it turn out to be useful.

91

cosh2(

2 |F |12 y

)

− sin2 θ

2(5.68)

which never becomes negative. What we should realize is the fact that both hyperbolic cosineson the l.h.s. of (5.68) are conjugated quantities, so its product is bound to be real and non-negative, and we can apply the saddle-point technique

limN→∞

1

NlnZ (J, B) = ln 2+

+ limN→∞

1

Nln

∫ ∞

−∞

[

e−y2+ 1

2ln

h

cosh2“

2|F |12 y

”

−sin2 θ22

i]N

dy. (5.69)

The maximum of

g (y) = −y2 +1

2ln

[

cosh2(

2 |F |12 y

)

− sin2 θ

2

]

(5.70)

gives us the saddle-point equations

−y0 +|F |

12

2

sinh(

4 |F |12 y0

)

cosh2(

2 |F |12 y0

)

− sin2 θ2

= 0, (5.71)

−1 + 2 |F |cos2 θ

2 cosh(

4 |F |12 y0

)

− sinh2(

2 |F |12 y0

)

cosh2(

2 |F |12 y0

)

− sin2 θ2

< 0. (5.72)

Thus, the free energy is

f (F, h) = ln 2 + g (y0) (5.73)

where y0 verifies (5.71). The ghost variable y is related to mS , the staggered magnetization.This relationship is obscure, in the sense that, in order to see this, we have to go back to (5.67),avoid the transformation (5.68) and set different magnetic fields θ1 and θ2

〈mj〉j=1,2 = mj =∂f

∂iθj

2

=∂g

∂iθj

2

∣∣∣∣∣y=y0

+∂g

∂y

∣∣∣∣y=y0

∂y

∂iθj

2

=∂g

∂iθj

2

∣∣∣∣∣y=y0

(5.74)

for the saddle-point equation forces ∂g∂y

∣∣∣y=y0

to be zero.

Once m1 and m2 are well known, the calculation of m and mS is straighforward. However,if the equality θ1 = θ2 does not hold, the saddle point equation becomes complex again, so it isa bold step to consider the new function g (y) as the free energy. Nevertheless, we regard thisparticular steps as an operational trick; since the final result should be evaluated at a y obtainedwith the true saddle-point equations (5.71), there is nothing wrong with this approach. Aftersome calculus,

92

m1 = 12

cosh“

2|F |12 y0

”

sinh“

2|F |12 y0

”

+i sin θ2

cos θ2

cosh2“

2|F |12 y0

”

−sin2 θ2

, (5.75)

m2 = −12

cosh“

2|F |12 y0

”

sinh“

2|F |12 y0

”

−i sin θ2

cos θ2

cosh2“

2|F |12 y0

”

−sin2 θ2

, (5.76)

m =i sin θ

2cos θ

2

cosh2“

2|F |12 y0

”

−sin2 θ2

, (5.77)

mS =cosh

“

2|F |12 y0

”

sinh“

2|F |12 y0

”

cosh2“

2|F |12 y0

”

−sin2 θ2

. (5.78)

Therefore, and using (5.71),

y0 = |F |12 〈mS〉. (5.79)

This results are quite interesting: the magnetizations of the individual sublattices are, in general,complex, and they are related to each other by conjugation,

m1 = −m∗2.

On the other hand, the staggered magnetization

mS = m1 − m2 = 2Re (m1)

is always real, and by virtue of (5.79), the solution y0 to the saddle-point equation is real aswell. The magnetization m becomes purely imaginary

m = m1 + m2 = 2iIm (m1)

in the presence of a θ term. For real external fields, the solution obtained here becomes thestandard mean-field solution to the antiferromagnetic Ising model.

The resulting mean-field equations are quite similar to the standard ones (showed in ap-pendix C), but the antiferromagnetic nature of the system makes them a bit more complicated31.

m1 = 14

i sin θ+sinh(4|F |mS)

cosh2(2|F |mS)−sin2 θ2

, (5.80)

m2 = 14

i sin θ−sinh(4|F |mS)

cosh2(2|F |mS)−sin2 θ2

, (5.81)

m = 12

i sin θcosh2(2|F |mS)−sin2 θ

2

, (5.82)

mS = 12

sinh(4|F |mS)

cosh2(2|F |mS)−sin2 θ2

. (5.83)

The phase diagram in the mean-field approximation

All the magnitudes we have computed depend explicitly on the staggered magnetization mS ,which in turn depends on F and θ. Therefore the saddle-point equation (5.71), or its equivalent(5.83), must be solved first, in order to calculate the dependence of the other observables withθ. However, these are implicit equations, and in order to solve them we must rely on numerical

31In fact, another different set of mean-field equations were proposed to this system as an ansatz in [96]. Fora discussion on the validity of these equations, see the Appendix C

93

methods. A simple way to see the solutions of (5.83) is a plot where the l.h.s. and the r.h.s.of (5.83) are represented as a function of mS . The solutions are the coincident points of bothfunctions. In 5.23, there is only one solution mS = 0, which is a maximum of the free energy,but for other values of F the situation changes dramatically, and two new symetric solutionsappear, which are maxima of the free energy, and the original solution mS = 0 becomes aminimum, and it is no longer relevant.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

mS

Paramagnetic phaseCritical line

Antiferromagnetic phasemS

Solutions

×

×

×

×

Figure 5.23: Plot done for θ = 0 and three different F values corresponding to the two differentphases plus the critical Fc. A solution to the saddle point equations can be found at thosepoints marked by the circles.

From the two saddle point equations, (5.71) and (5.72), the critical value of F , Fc can beascertained, and it depends on θ as

2Fc = cos2θc

2. (5.84)

Consequently, for a given value of θ, there are values of F that make the system acquire anon-zero staggered magnetization. This last equation gives us the phase diagram on the F − θplane

and there is a second order phase transition at the critical line (5.84). The behaviour of thesystem in the paramagnetic phase (given by mS = 0 and 2F < cos2 θ

2) is the same as theLangevin theory, and the magnetization m (5.82) equals that of (5.58) after the proper substi-tution h → iθ. As θ → π the paramagnetic phase narrows, until it is reduced to a single point

F = 0 at θ = π. The staggered phase (with antiparallel spin ordering mS 6= 0 and F >cos2 θ

2

2 )on the other hand features Z2 spontaneous symmetry breaking at θ = π, as equation (5.83)indicates. The fact that this model features a phase transition at non-zero values of the externalfield is quite remarkable indeed. This kind of transitions would never appear in a ferromag-netic model, as the external field and the spin coupling work in the same direction: parallelspin alignment. On the contrary, in the antiferromagnetic case, the introduction of an externalfield produces frustration, whose origin comes from the competition of the spin coupling, try-ing to move the spins towards an antiparallel configuration, and the external field, favouring acompletely parallel structure.

All the magnetizations are continuous functions, but the staggered susceptibility χS diverges,

94

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0π4

π2

3π4

π

F

θ

Critical Line

Figure 5.24: Phase diagram of the mean-field approximation to the antiferromagnetic Isingmodel in the F − θ plane.

as usually happens in a second order phase transition. The topological32 susceptibility χT , onthe other hand, displays a gap at the critical line. The computation is tedious, and has beenmoved to Appendix D. I only show here the final result

∆χT = limθ→θ+

c

χT − limθ→θ−c

χT =3

4 |F |2 |F | − 1

4 |F | − 3. (5.85)

Finally, the critical exponent γλ for this mean-field theory can be calculated, to see if it co-incides with that obtained in simulations. In order to do so, we expand m in the neighbourhoodof θ = π

m (θ) ∼ m (π) +∂m

∂θ

∣∣∣∣θ=π

(π − θ) +∂2m

∂θ2

∣∣∣∣θ=π

(π − θ)2 + . . . (5.86)

If γλ is not natural number, we expect the first non-zero derivative to diverge. On the contrary,if γλ is a natural number, the order of the first non-vanishing derivative will give us the criticalexponent. Taking derivatives

∂m

∂θ

∣∣∣∣θ=π

=i

2

(2 cos2 θ − 1

)cosh2 (2 |F |mS) + 1 − cos2 θ

2(cosh2 (2 |F |mS) − sin2 θ

2

)2 −

− i

2

2 |F | sin θ sinh (4 |F |mS) dmS

dθ

∣∣∣θ=π

(cosh2 (2 |F |mS) − sin2 θ

2

)2

∣∣∣∣∣∣θ=π

=

= − i

2

1

sinh2 (2 |F |mS)+

2 |F | sin θ sinh (4 |F |mS) dmS

dθ

∣∣∣θ=π

sinh4 (2 |F |mS)

(5.87)

The first term on the r.h.s. of (5.87) does not diverge if mS is not vanishing as θ → π. AsFc → 0 in this limit, we expect a non-zero value for mS if F is non-zero. Hence this term takesa finite, non-vanishing value. The next term is proportional to

32Topological in the sense that M/2 is a quantized charge, m/2 is its associated charge density, and χT thesusceptibility.

95

limθ→π

dmS

dθsin θ.

If dmS

dθ diverges, then its product with sin θ may yield a non-zero contribution to the final criticalexponent. However, if it stays finite, the sine function kills this term at θ = π. After a tediouscalculation, it can be shown that the derivative vanishes, therefore

m (θ) ∼ iπ − θ

2 sinh (2F |mS |)= K (π − θ) , (5.88)

with K a non-zero constant. The magnetization behaves as (π − θ)γλ−1 in the neighbourhoodof θ = π. Thence, for mean-field antiferromagnetic theory γλ = 2, and the symmetry is alwaysbroken for F 6= 0. For F = 0 the behaviour is that of the Langevin theory, and γλ = 0.

Since mean-field theory works better in high dimensional systems (it reproduces all thecritical exponents exactly for the ferromagnetic Ising model in dimension 4 and above), andthe exponent γλ seems to have settled in γλ = 2 for the two- and three-dimensional models,and for the mean-field approximation, we expect this result to hold for any dimension of thesystem. This is not a proof, but in fact, it would be very remarkable if the behaviour of theantiferromagnetic Ising model in a higher dimension departed from γλ = 2.

5.5 Conclusions

Although the aim behind this investigation of the antiferromagnetic Ising model is to test ourtechniques for a future application to QCD in presence of a θ term, the results obtained throughthis chapter deserve attention on their own merit. Using the method described in section 5.3,the order parameter for the Z2 symmetry can be calculated for any value of θ, and althoughthere are some regions of the phase diagram where the method does not work very well, itprovided us with enough information to make an educated guess on the phase diagram of thetheory.

Our guess was later confirmed by a long mean-field calculation, which shares many propertieswith the original model. The mean-field result is quite interesting by itself, for it can becompletely solved. The results of [94] and [95] supplied the remaining information for the two-dimensional case. In the end, we were able to reconstruct qualitatively the whole phase diagramof the theory for two-dimensions, and although we did not pursue to solve the model for higherdimensions, the mean-field calculations, and the fact that the behaviour for the two- and thethree-dimensional models is the same for large values of F , give us strong indications that thisphase diagram is qualitatively valid for any dimension of the model.

The method only has two caveats: (i.) it does not work properly (and can give wrongresults) if there is a phase transition for some θ < π, and (ii.) for small values of the couplingF the required extrapolations are not feasible. Fortunately for us, (i.) the standard wisdom onQCD, based on reasonable assumptions, expects no phase transitions for θ < π, and (ii.) QCDis an asymptotic free theory, thus its continuum limit lies in the region where the extrapolationsof the method work well. Therefore this method has become the perfect candidate to performsimulations of QCD with a θ term, which might provide the scientific community with preciousinformation for the understanding of the strong CP problem.

96

Part II

Polymers and the chemical potential

97

Chapter 6

QCD and the chemical potential

“It is characteristic of wisdom not to dodesperate things.”

—Henry David Thoreau

Nowadays, the scientific community acknowledges the existence of new states of the hadronicmatter at temperatures or densities higher than the energy scale of hadrons. In such conditions,the wave function of the quarks composing the nuclear matter begin to overlap notoriously, andthe distance among quarks becomes small enough that the individual hadronic and mesonicparticles are indistinguishible. Asymptotic freedom for QCD tell us that this new state ofmatter should behave as a free ideal gas of quarks and gluons, the so called quark-gluon plasma(QGP ). It is also believed that in these regimes, the quark masses becomes irrelevant, andchiral symmetry is restored. Besides this well established phase, for very high densities andlow temperatures, a color-superconducting phase transition is expected to take place: The largefermi spheres become unstable as the attractive force weakens, due to the asymptotic freedom,resulting in quark-pairing near the Fermi surface and a non-vanishing diquark condensate 〈ψψ〉.Both states of quark matter are not very common in our everyday-life, nonetheless they are ofparamount importance to understand the behaviour of universe in its earlier stages. Moreover,the description of the complete phase diagram of QCD under these extreme conditions wouldhelp to understand the composition of many astrophysical objects, like neutron stars. But theapplication of these studies is not restricted to such entities; the heavy-ion collisions generatedat large facilities like RHIC and LHC are expected to be described accurately by assumingthat the matter involved behaves as a QGP at high temperature and density, and then freezesinto hadrons.

Of both fields (study of QCD at non-zero temperature and at non-zero densities), finitetemperature QCD is far more developed. Lattice QCD simulations of finite temperature sys-tems have been able to predict the critical temperature TC ≈ 200 − 250 MeV at which: (i.)the transition hadronic matter-QGP occurs, and (ii.) chiral symmetry is restored, and it seemsthat these two phenomena are inherently linked. On the other hand, finite temperature latticecalculations are used to predict or explain the data obtained in the heavy-ion colliders. It isclear that finite temperature QCD is an evolving field.

Sadly, and in spite of the community efforts, the same praise can not be given to finitedensity QCD. Direct lattice simulations are hindered by the well known sign problem, andevery attempt to solve it ends in frustration. It seems that the state of the art of finite densityQCD is almost the same than three decades ago, despite the continuous research performedduring the last years.

99

But preseverance is rewarded, and step by step, new algorithms appear that seem to work,at least partially, and they can give us new insights on the behaviour of the matter at veryhigh densities. The polymeric approach, or the complex Langevin are two examples of half-succeeding algorithms which, even if they fail to solve the problem completely, at least theykeep the hope of finding the right solution one day.

6.1 Introducing the chemical potential on the lattice

The usual way to introduce new features on the lattice is to generalize continuum procedures orprescriptions to a discrete space-time. As we know very well the theories in the continuum, thismethod is often attempted in first place. Nevertheless, the results yielded by this schema are notalways right. One clear example is the discretization of fermions on the lattice, leading to thewell known problem of the doublers and the anomaly [8]. Another example is the introductionof a chemical potential on the lattice.

Naively, we expect the chemical potential to be coupled to the current associated with thefermion number ψγ4ψ, so the lagrangian density of the theory is modified as

L → L− aµψγ4ψ. (6.1)

Then, we can calculate the µ-dependence of the energy density ǫ taking into account thenew partition function Z,

ǫ = − 1

V

∂Z

∂β

∣∣∣∣βµ=Fixed

, (6.2)

where Z is written as a function of the new lagrangian density

Z =

∫∏

x

dψx dψxeR

d4xL. (6.3)

In the case of free fermions, the calculation for naive fermions1 can be carried out exactly,

ǫ =( µ

2a

)2∫ π

−πd4p

1(∑3

j=0 sin2 pj + (ma)2)2

− 2iµa sin q4 + 4µ2a2

, (6.4)

and this quantity diverges as ǫ ∼( µ

2a

)2in the continuum limit a → 0, which is in clear contrast

with the continuum result for massless fermions ǫ ∼ µ4.This divergence is not a lattice artifact. In fact, the continuum calculation leads to the same

divergence at some point, but there, a prescription is used to remove it. We could proceed inthe lattices as in the continuum, and add counterterms to get rid of the divergence [97], butthis is, a priori, a very awkward way to proceed, that might lead to cumbersome formulationsof finite density theories2. It is desirable to find a more elegant description of the chemicalpotential on the lattice.

The first successful attempt to circumvent this problem must be credited to P. Hasenfraztand F. Karsch [98]. They observed that the chemical potential in the euclidean formulation ofthermodynamics acted as the fourth component of an imaginary, constant vector potential, andthat the standard way of introducing the chemical potential leads to gauge symmetry breaking.

1The inclusion of any other kind of fermions does not modify the final dependency. In fact, the problemalways arises, regardless of the regularization. I choose here to show the results for naive fermions because oftheir simplicity.

2The formulation exposed in [97] however turned out to be quite manageable.

100

The clearest example is QED in the continuum, where the chemical potential is introduced asa shift µ/e in the time-component of the vector potential A, which can be thought as a gaugetransformation3.

This way, for naive fermions, the fermionic matrix looks like

∆xy = mδxy +1

2

∑

ν=1,2,3

{

γν

[

Ux,νδy,x+ν − U †x−ν,νδy,x−ν

]

+

1

2γ4

[

eaµUx,4δy,x+4 − e−aµU †x−4,4δy,x−4

]}

, (6.5)

Using this prescription,

ǫ =1

8π3a3

∫ π

−πd4pθ

(

eµ − K −√

K2 + 1) K√

K2 + 1,

K =3∑

i=1

sin2 pi, (6.6)

with θ(x) the heaviside step function. The continuum result for the momentum cut-off θ(

µ −√

p2 + m2)

is recovered in the a → 0 limit4.

Unfortunately, neither the fermionic matrix (6.5), nor any of its variations (Kogut-Susskind,Wilson or Ginsparg-Wilson fermions, to give some examples) are suitable for numerical simu-lation. The reason is that hermiticity is lost for the Dirac operator. Then, any power of thedeterminant of the fermionic matrix becomes complex, and the sign problem appears. In thecase of Wilson fermions, hermiticity is violated beforehand, even at zero chemical potential, butthe following relation holds

γ5D = D†γ5 =⇒ γ5D = (γ5D)† , (6.7)

so Det [γ5D] is real and although it can be negative, simulations can be performed for an evennumber of degenerated flavours, for Det2 [γ5D] is always positive.

The problem of the chemical potential is that it breaks completely the relation (6.7) byintroducing the asymmetry between forward and backward links. This asymmetry is necessary,for it enhances the propagation of quarks and hinders that of antiquarks, but indeed it is notof any help when it comes to do the lattice simulations.

In fact, during the 90’s [100, 101, 102, 103], several groups tried to perform simulationsat finite density by means of the quenched approximation, but they obtained results in clearcontradiction with the intuitive expectations: It is expected that the observables be almost µ-independent as long as µ is smaller than a certain threshold value, given by µc = mB/3, wheremB is the lightest baryon mass. For µ > µc, a finite baryon density populates the system, as itis more convenient from the energetical point of view. But the quenched approximation set thecritical value of the chemical potential at m2

π/2, with mπ the mass of the pion. In the chirallimit, the baryons stay massive, whereas the pions become massless. Then µc is set to zero.This is completely inacceptable.

3The physical effect of the chemical potential in QED is irrelevant, as the introduction of µ amounts to aconstant shift in the energy. In general, any U (N) theory lacks baryons, and therefore, do not truly depend onµ.

4R. V. Gavai wrote a beautiful paper [99] where he shows a general action leading to the correct implementationof the chemical potential. Sadly it seems that there are no realizations of this general action with a real andpositive determinant.

101

A common explanation for this behaviour is to understand the quenched approximation asthe theory of the modulus (the one that solves the sign problem taking the modulus of thedeterminant, this theory treats equally the quark 3 and the antiquark 3∗ representations) forzero flavours [104]. As the theory of the modulus features a diquark baryon, the critical valuefor the chemical potential is expected to be around m2

π/2. Nonetheless, we can arrive to thequenched approximation from very different limits for instant, the zero flavour theory of fullQCD, and in this case the quarks and the antiquarks are treated in a different way, so theremust be other explanation for this behaviour. In fact, the differences between the theory ofthe modulus, full QCD and the quenched approximation has been investigated in the largequark mass limit [105], and for vanishing temperature, full QCD and the theory of the moduluscoincide, whereas the quenched approximation gives very different results.

The conclusion is that the sea-quarks are of paramount importance in finite density calcu-lations, and they can not be neglected happily, as the quenched approximation does. In thebeginning era of lattice QCD, where no much computer power was available, this was trans-lated in the necessity of developing competent algorithms to simulate fermionic actions: LatticeQCD simulations were commonly performed without dynamical fermions, either in its puregauge form, or including the fermions using the quenched approximation. Except for a few shyattempts, most of the lattice practitioners thought there were no satisfactory algorithms to dealwith the problem of the Grassman variables in a computer, and some of them were activelyresearching new computational methods for fermionic degrees of freedom.

6.2 Polymeric models

The problem which arises with fermions is how to make them comply with Pauli’s exclusionprinciple inside the lattice. Although theoretically a representation of anticommuting variablescould be set up in the computer, this is not feasible in practice [19]. Thence, we should removeour Grassman variables from our theory, thus we integrate them out to yield a determinant.But the determinant is a non-local quantity, which render the simulations slow, and in somecases (like finite density) is intractable, due to the sign problem.

The first steps

One interesting attempt to circunvent these problems was the polymer representation of fermions.The polymerization of fermions, although young, was quite promising, and had been appliedsuccessfully to some models [106, 107, 108]. It provided a method to simulate fermions withoutcomputing the fermionic determinant, and this property was quite appealing, for it would speedup the fermionic simulations by a large factor. However, there was a key handicap: A severesign problem appeared where it has never been. The polymerization transformed the fermionicdeterminant into fermionic loops, whose sign depended on the gauge group and on their partic-ular shape. In fact, the pioneers of polymeric simulations [106] did notice the abrupt decreasingof the mean sign, but they did not worry too much about it, as long as their numerical resultswere consistent with the expected results. QCD performs no better, and the simplest negativefermionic loop is very simple indeed [109] (see fig. 6.1), causing the negative configurations tobe almost as usual as the positive ones. As a result, the observables fluctuate wildly, and areimpossible to measure reliably. Some clustering techniques were introduced to avoid the signflipping, but these rendered the algorithm as slow as the standard determinant computation,thus the idea was forsaken.

P. Rossi and and U. Wolff recovered it in a shy attempt to applicate the polymerization tothe strong coupling limit of U(N) gauge theories with staggered fermions [107]. They observed

102

a) b) c) d)

Figure 6.1: Example of loops with positive and negative weights (assuming anti-periodic bound-ary conditions in the y direction). The loops a) and b) are the simplest of their kind, and carrya positive weight. On the other hand, the loops c) and d) contribute with negative terms to thepartition function.

that the family of gauge groups U(N) were not affected by the sign problem, since they lackedbaryonic loops5. They used this advantage to establish a relation between the chiral symmetrybreaking and confinement. Later on, U. Wolff introduced the chemical potential for a SU(2)gauge theory in this polymeric model [110]. Baryonic loops appeared, but they were alwayspositive, due to the nature of the SU(2) gauge group. As the fermionic loops were hard toimplement, he decided to add an external source, and compute the quantities in the vanishingsource limit.

Karsh and Mutter took the baton from Wolff and faced the SU(3) gauge group (QCD)[109], introducing a key modification: By cleverly clustering the polymers and the baryonicloops, they found a way not only to create baryonic loops easily by simulating only polymerdegrees of freedom, but also to solve the sign problem at zero chemical potential. It was possibleto simulate a non-zero chemical potential in a wide range of µ’s as well, and they found a firstorder phase transition between two clear different states, at a critical µ which was very close tothe mean field predictions. The Monomer-Dimer-Polymer model (MDP ) was born.

Fall and rise of the MDP model

Unfortunately, there was something wrong with Karsch and Mutter method. A forecomingarticle by Azcoiti et al [111] proved that Karsch and Mutter computer implementation of thepolymer theory (which was that of Rossi and Wolff) featured a poor ergodicity when baryonicloops were added, and therefore, their simulations where no longer reliable. Being more specific,it was impossible to simulate a lattice whose temporal length (LT ) was higher than four. In thosecases, the system would freeze in one of the two states, regardeless of the chemical potentialvalue, resulting in a inacceptably large hysteresis cycle. Moreover, the algorithm was extremelyinefficient when dealing with low masses, and even Karsch and Mutter reckoned in their articlethat ergodicity could not be proved. The idea, again, was abandoned.

5A derivation of this statement will be done later.

103

Recently, Ph. de Forcrand and collaborators rescued the Karsch and Mutter polymerizationof the action, and substituted their naive algorithm by a world-line approach [112]. The world-line algorithms were first brought to the scientific mainstream by N. Prokof’ev and B. Svistunov[113, 114, 115] long time ago, but they didn’t draw too much attention. Chandrasekaran tookthem back to life in his later works [116], arguing that this kind of algorithms may solve thesign problem for certain actions. The first results of Ph. de Forcrand were both promising anddeceptive. On the bright side, they broke the zero mass limit, and the simulations performedwell. On the dark side, they never showed results beyond LT > 2, so the low temperature (andthe most interesting) region of the phase diagram was never explored. Later on, they pushedthe worm algorithm trying larger lattices in temporal extent and playing with the temperature,obtaining some remarkable results [117]. It seems that different algorithms yield different results,and work well in different ranges. This leads us to the conclusion that, there is nothing wrongwith Karsch and Mutter way to rewrite the action, or the partition function. The problem isrelated to the simulation algorithm.

Starting from Karsch and Mutter algorithm, both ergodicity and efficiency can be improvedby adding new possibilities of interaction among polymers, even at zero mass. These modifica-tions allow us to simulate strong coupling QCD with any temporal length. Our conclusion isthat the Karsch and Mutter regrouping of configurations does not solve the sign problem forthe chemical potential, although there are some interesting regions where simulations can beperformed.

6.3 Karsch and Mutter original model

The starting point of the polymeric formulation of strong coupling QCD is the partition functionincluding both, gauge and fermion fields

Z (m, µ, β) =

∫

[dU ] dψ dψeSGeSF (6.8)

where ψ and ψ are, as usual, the fermionic Grassman variables, and U belongs to the gaugegroup (SU(3) for QCD). The gauge action SG is no longer relevant, for we are in the strongcoupling regime (β → 0). The fermionic action, on the other hand, is responsible for thecomplete dynamics of the system. We choose, as P. Rossi and U. Wolff did [107], the staggeredformulation of fermions, although we are aware of the fact that the polymerization of Wilsonfermions is not an untreated problem [118, 119]. The fermion action reads

SF (β) =∑

n,ν

[ξn,ν(µ)

(ψnUn,νψn+ν − ψn+νUn+ν,−νψn

)+ 2mψψ

]

= ψm∆mnψn (6.9)

where the factor ξn,ν(µ),

ξn,ν(µ) = ην(n)

eµ ν = 4e−µ ν = −41 otherwise

(6.10)

encloses both the staggered sign

ηµ(n) = (−1)P

i<µ xi , (6.11)

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which depends on the point n, the link direction ν and the chemical potential factor eµ. Weare considering ν = ±1,±2,±3,±4 in 3 + 1 dimensions, being ν = ±4 the positive (negative)temporal direction.

As we are in the strong coupling regime, the direct integration of the partition functionshould be feasible. Of course we know that the fermionic action reduces to the determinant ofthe fermionic matrix

Z (m, µ, β = 0) =

∫

[dU ] det∆mn, (6.12)

but we are interested in integrating the gauge fields as well, so let’s go back one step and tryanother approach. We expand the exponential of the fermion action

eSF =∑

i=0

SiF

i!, (6.13)

and then integrate the gauge fields usign the group integration rules. The big problem herearises from the powers of SF . The fermionic action (6.9) is a sum over all possible points nand directions ν, and any power of this large sum involves an even larger number of sumands.What we want to do is to integrate all these sumands, one by one.

Let’s perform the integration step by step. First of all, we expand the mass term.

Jn (m) = e2mPN

k=1 ψknψk

n =∑

i=0

(

2m∑N

k=1 ψknψk

n

)i

i!. (6.14)

where k stands for a color index. We have called this term Jn (m), for further reference.Following the standard notation on the subject,

Mk(n) = ψknψk

n, M(n) =∑

k

ψknψk

n, (6.15)

and we will call a monomer to the first one, Mk, for any k. The expansion (for QCD, N = 3)in terms of these new fields looks like

Jn (m) = 1 + 2m (M1(n) + M2(n) + M3(n))

+(2m)2 (M1(n)M2(n) + M2(n)M3(n) + M3(n)M1(n))

+(2m)3M1(n)M2(n)M3(n). (6.16)

Monomers are discrete, ultralocal objects which consume some Grassman variables of thepoint n they belong to, in particular the pair ψk

nψn for a given value of the color k. For SU(N),the monomer occupation of a site number is an integer whose maximum value is N , the numberof colors. In this case (maximum monomer occupation number), all the Grassman variables ofthe point are used to construct monomers. The other extreme case is zero occupation number,and the Grassman variables may serve other purposes.

These other purposes are related to the terms which involve gauge fields. The integration ofthese turns out to be fairly easy, for in the strong coupling regime only the annihilating fieldssurvive. For any U(N) gauge group, this set of annihilating fields reduce to the product of a

link Un,ν and its adjoint U †n,ν = Un+ν,−ν , which equals unity, but in the special case of SU(N),

the group integral of the N th power of a field is also the identity, so the combination UNn,ν should

be taken into account as well.

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After the gauge integration, only a sum of different products of Grassman variables remain,and we must assure that every Grassman variable appears once and only once in each sumand,otherwise the corresponding summand becomes zero. Translating the result into monomeroperators (6.15), what we find for QCD is

In,ν (µ) =

∫

[dU ] exp[ξn,ν(µ)

(ψnUn,νψn+ν − ψn+νUn+ν,−νψn

)]

= 1 +1

3M(n)M(n + ν) +

1

12(M(n)M(n + ν))2 +

1

36(M(n)M(n + ν))3

+ξ3n,ν(µ)B(n)B(n + ν) + ξ3

n+ν,−ν(µ)B(n + ν)B(n). (6.17)

The quantities

B(n) = ψ1(n)ψ2(n)ψ3(n) B(n) = ψ1(n)ψ2(n)ψ3(n) (6.18)

can be regarded as baryon-antibaryon fields. These fields come from the cubic terms(ψnUn,νψn+ν

)3,

and do not appear in the U(N) gauge groups. As we shall see later, these fields are related tobaryons, so an interesting conclusion we anticipated before is the fact that U(N) gauge groupsdo not allow the creation of baryons6.

Following Karsch and Mutter program, and in order to complete the interpretation of the re-sulting partition function, we translate the product of monomers M(x)M(n+ν) into dimer fieldsDi(n, ν). A dimer field arises from the cancelation of the gauge fields in the terms ψnUn,νψn+ν

and ψn+νUn+ν,−νψn. This cancellation gives way to the Grassman product ψnψn+νψn+νψn, aproduct of Grassman variables which involve two points joined by a link Un,ν ; so the dimers arenot ultralocal entities, as the monomers, but they link two neighbouring points.

The double dimer field

D2(n, ν) =

(M(n)M(n + ν)

2!

)2

(6.19)

can be written as

D2(n, ν) = [M1(n)M2(n) + M1(n)M3(n) + M2(n)M3(n)]

× [M1(n + ν)M2(n + ν) + M1(n + ν)M3(n + ν)+

M2(n + ν)M3(n + ν)] , (6.20)

for the other Grassman products involve powers higher than one of at least one of the Grass-man variables, and therefore, they cancel after integration. The factorials are introduced forconvenience. The triple dimer field is easier to build

D3(n, ν) =

(M(n)M(n + ν)

)3

(3!)2

= M1(n)M2(n)M3(n)M1(n + ν)M2(n + ν)M3(n + ν) (6.21)

as there is only one possibility to make use of all the Grassman variables. On the other hand,the one dimer field involve nine terms in the SU(3) formulation

D1(n, ν) = M(n)M(n + ν) = [M1(n) + M2(n) + M3(n)]

6Another way to visualize the lack of baryons in U (N) groups is to notice that a baryon would not be a gaugeinvariant object.

106

× [M1(n + ν) + M2(n + ν) + M3(n + ν)] (6.22)

Let’s put these new fields in the integral In,ν (µ) (6.17)

In,ν (µ) = 1 +1

3D1(n, ν) +

1

3D2(n, ν) + D3(n, ν)

+ξ3n,ν(µ)B(n)B(n + ν) + ξ3

n+ν,−ν(µ)B(n + ν)B(n). (6.23)

Note that there is a ξ3 factor weighting the baryonic terms, which can become negative. Thethird power comes from the number of colors N = 3; which means that for N even, there is nosign problem for the baryons in this formulation. This is not a great surprise: The sign problemis intrinsecally related to fermi statistics, but the baryons of SU(N) for N even are bosons.

In the end, the partition function is written as a product of dimer, monomer and baryonicfields:

ZMDP (m, µ, β = 0) =∑

n,ν

∫

dψ dψ In,ν (µ) Jn (m) . (6.24)

The only terms counting in this integral are those which make use of all the Grassman variablesin each point, in one way or another. This fact leave us a simple rule: The number of monomers(nM) and outgoing links (nD, number of dimers) in any point of the lattice must equal thenumber of colors N

nM(n) +∑

ν

nD(n, ν) = N (6.25)

For QCD, N = 3, and there is a finite set of point types, shown in Table 6.1. The factor w(x)appearing in the last row is a statistical weight derived from the integrals Jn (m) and In,ν (µ).In simple words, the number w(x) takes into account the colour-multiplicity allowed for eachlink, except for the 1/3 factor of equation (6.23), associated to single and double dimers; thishas to be taken into account separately.

Node type 0 1 2 3 4 5 6

nD(x) 3 2 1 0 2 3 3nM (x) 0 1 2 3 1 0 0w(x) 3 6 3 1 3 6 1

Table 6.1: Allowed node types for the SU(3) theory.

The later rule (6.25) is violated in presence of baryonic loops, for these entities consume allthe Grassman variables available, so there is no place for a monomer or a dimer. Let’s see why:A baryonic loop B(n)B(n + ν) uses the ψk of all colors (k = 1 . . . N) of a point n, and the ψk

of its neighbour (n + ν). In order to kill the N Grassman fields ψ that dwell in the point n, weneed to put there N Grassman fields ψ, i.e., we have to close the baryonic loop at some point.

On the whole: We have rewritten the partition function of a strong coupling gauge SU(N)theory by integrating both, Grassman and gauge fields. As a result, we have obtained a discrete

107

set of configurations consisting on graphs of monomers, dimers and baryonic loops. Then weperform a Montecarlo simulation in the space of graphs.

So far we have described Rossi and Wolff’s modelization of strong coupling U(N) andSU(N). In their first article, Rossi and Wolff restricted their simulations to U(N) gauge groups,for it is very hard to implement baryonic loops. As we have explained before, whenever a bary-onic loop populates a site, this site becomes saturated, and no other structure (monomer ordimer) can be there. Given a configuration with monomers and dimers, the addition of a bary-onic loop involves a non-trivial change in all the neighbouring sites of the loop, and this changemay extend to even further sites, in order to preserve the rule (6.25), rendering the algorithmhighly non–local.

But Wolff developed a way to include baryonic loops, and tested it in SU(2) [110]. He addedto In,ν (µ) an external source like this

In,ν (µ)W = In,ν (µ) + λB(n) + λB(n). (6.26)

We recover the standard action in the limit λ → 0. The external source allows the baryonicloops to be open, for now their endings are annihilated by the external source terms. Butthere may be a severe sign problem in this formulation, for the open baryonic chains acquirearbitrary sign, due to the ξN Kogut-Susskind weights. This problem can be eliminated in SU(2)easily: Being all the Kogut-Susskind weights squared, we can forget about the sign of the openchains. Nevertheless, it is impossible to implement SU(3) this way, and the sign problemhinders any attempt of simulation. Moreover, the dynamical creation of baryonic loops relieson the interaction of the open baryonic chains, generated by the external baryonic field. Asthe external field is removed (which is the interesting limit), these open chain interactions arereduced as well, slowing down the algorithm. This does not seem like the best way to introducethe baryonic loops on the lattice.

The solution came from the hands and the minds of Karsch and Mutter. They realized thatthe points of type zero (see table 6.1) could be arranged in closed loops (polymers), as shown infig. 6.2. These polymers consumed all the Grassman variables available along the loop, so theybehaved like a baryonic loop. By associating a baryonic loop to each polymer, they found away to avoid this non-locality in the generation of configurations procedure. The key point hereis the clustering of two configurations into one: the polymer configuration A and the baryonicloop configuration B. This new configuration has a weight equivalent to the sum of the weightsof A and B, and we can create polymers easily, by proposing local changes.

In addition there is a further advantage, let’s compute the weight of a baryonic loop C in aconfiguration, being C the set of points and directions that result in a closed loop. There aretwo main possibilities for C: It may wind around the temporal direction of the lattice or not.Since the net temporal displacement is zero in the later case (we will assign the subindex SL,spatial loop to it), the accumulating eµ factors cancel the e−µ, and only a product of staggeredsign factors remain, regardeless of the orientation of the loop,

wSL =∏

n,ν∈C

η3ν(n) =

∏

n,ν∈C

ην(n). (6.27)

On the other hand, a winding loop (or TL, temporal loop) keeps the exponential factors, so doesits weight, and the final result is

w±TL =

∏

n,ν∈C

η3ν(n)e±3µ = −(−1)ke±3µkLT

∏

n,ν∈C

ην(n). (6.28)

where k is the winding number. The factor (−1)k arises from the antiperiodic boundary con-ditions in the temporal coordinate. The exponent ±3µkLT is positive in the case of a baryon,

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A B

Figure 6.2: Configuration A shows a temporal polymer, winding around the lattice through theboundary, and a spatial polymer. In configuration B the fermionic loops equivalent to the poly-mers of configuration A are shown. In the Karsch and Mutter model, these two configurationsare the same, and we have to add their weights in order to calculate its contribution to thepartition function.

and negative in the case of an antibaryon. Clearly, the simulation of baryonic loops ”out ofthe box” features a sign problem, even at zero chemical potential. The configuration weight isnegative whenever there exists an odd number of negative loops, and given the point weight (inTable 6.1), this possibility seems rather plausible.

Fortunately, the clustering method of Karsch and Mutter solves the problem for µ = 0.Adding weights of a spatial loop and a polymer results in

wSL + wPoly = 1 +∏

n,ν∈C

η3ν(n) =

{2 if wSL = 10 if wSL = −1

, (6.29)

and the negative spatial loops do never appear, as they have vanishing weight. We can dis-tinguish the orientation of the loop looking at the orientation of the polymer, but there is noadvantage in doing so.

The case of a temporal loop is more subtle. First of all, Karsch and Mutter clustered thebaryon and the antibaryon loop into one entity, so we do not distinguish the orientation of theloop.

w+TL + w−

TL = −(−1)k2 cosh (3µkLT )∏

n,ν∈C

ην(n). (6.30)

Two polymer configurations are clustered with this one, so the final weight is

w+Poly + w−

Poly + w+TL + w−

TL = 2 − (−1)k2 cosh (3µkLT )∏

n,ν∈C

ην(n), (6.31)

which doubles that of the spatial loop in the µ = 0 case, for we have clustered the two possibleorientations of the loops. Now, Karsch and Mutter decide to distinguish between the differentpolymers, so they split this macroconfiguration into two different ones, each of these comprisinga polymer and half a baryon-antibaryon sum. The partition function is the same, but the weightnow is divided by two

109

Polymers Fermionic loops

Figure 6.3: The fermionic loops are oriented, so for a given closed path, there are two fermionicloops associated to it (right). Fortunately, the number of polymers associated to a closed path istwo as well (left), so we can even distinguish the orientation of the loops in our clustering of con-figurations. As explained in the text, Karsch and Mutter chose to cluster all the configurations,making the orientation of the loops irrelevant.

w±Poly +

w+TL + w−

TL

2= 1 − (−1)k cosh (3µkLT )

∏

n,ν∈C

ην(n) =

1 + σ (C) cosh (3µkLT ) . (6.32)

where σ (C) = ±1 gathers all the terms contributing to the sign of the loop.On the whole, the contribution of a given MDP configuration, labeled K, to the partition

function is

wK = (2m)NM

(1

3

)ND1+ND2 ∏

x

w(x)∏

C

w(C) (6.33)

where NM refers to the total number of monomers, NDj is the number of type j dimers (j = 1single dimer, j = 2 double dimer), w(x) is the specific weight of each type of point, and w(C) isthe weight of each loop present in the configuration. The equivalence with the original partitionfunction (before configuration clustering) is straightforward.

We have conserved most of the notation of Karsch and Mutter, as their paper [109] is thebase of this work.

Observables

The interesting observables in finite density QCD are those which signal a phase transition,namely the baryonic density and the chiral condensate.

i. The chiral condensate can be readily calculated using the expression

⟨ψψ

⟩=

1

V

d lnZ

d (2m)=

〈NM 〉2mV

, (6.34)

110

which equals the monomer density up to a normalization constant. Here V represents thetotal volume.

ii. The baryonic density is related to the baryonic loops, winding k times around the latticein the time direction

n =1

3VS

d lnZ

d (µ)=

sinh (3µkLT )

cosh (3µkLT ) + σ (C), (6.35)

and this quantity saturates as µ → ∞, for in that limit, coshx ≈ sinhx ≫ 1. The factor VS

factor represents the spatial volume only, and the 1/3 normalizes by the number of colors.

As there are negative weights in the partition function, we must evaluate these logarithms withcare. Actually, what we are doing is taking the modulus in the measure, and introducing thesign in the observables

〈O〉 =〈σKO〉+〈σK〉+

(6.36)

with σK the sign carried by the weight of the configuration K in the partition function sum.The symbol 〈〉+ indicates that we are using the modulus of the measure.

Implementation

After a random configuration of monomers and dimers was generated, the updating schema ofthe algorithm was quite simple: It only tried to increase or decrease the occupation numberof each site by transforming dimers connecting two sites x and y into two monomers, or twomonomers belonging to neighbouring sites x and y into a dimer. It is clear that this approachslows down critically when taking the chiral limit. The reason is simple: In order to reach theconfiguration B of fig. 6.4 from configuration A, the system is forced to create monomers. Butthe probability of creating a pair of monomers is proportional to the square of the quark massm. Thus, when close to the chiral limit, the system is doomed to stay in configuration A. Inother words, the acceptance of the original algorithm drops to zero in the chiral limit.

This rendered the algorithm useless for small masses. Even if the chemical potential isridiculously high, a system beginning in a completely random configuration never developsbaryonic loops, if the fermion mass is small enough, but freezes in a bizarre configuration ofdimers. Moreover, starting from a random configuration and simulating with a low mass, thesystem will be depleted of monomers soon. But this does not mean that all the monomers areeliminated. In fact, it might happen that, as the monomers are eliminated randomly, somemonomers become isolated, I mean, without other monomers in their neighbourhood. Thealgorithm can neither eliminate nor move these monomers if the mass is too low, leading topersistent dislocations and point defects in the lattice. This behaviour derives in undesireddefects in the lattice, that can not be eliminated.

The problem is accentuated with the cold configurations and the inclusion of fermionic loops,which are entities with great weights in the partition function. The only way to break a loopis to introduce a pair of monomers therein. Thanks to the m2 factor, as m → 0 the probabilityof breaking a loop becomes completely negligible, and the baryonic loops become superstableentities. This can lead to wrong results for the critical point, or to incredible large hysteresiscycles, as observed in [111].

Finally, the baryonic loops cannot interact among them in any way, unless they are firstbroken and then reconstructed. So two neighbouring loops of winding number k = 1 cannotbecome a loop of winding number k = 2 easily, and vice-versa. This fact implies that the loops

111

A B

P1 P2

P3 P4

P1 P2

P3 P4

Figure 6.4: Example of bottleneck in the algorithm proposed by Karsch and Mutter. Theprobability of transition A→B should be large O (1), for in B a new baryonic loop is created.Nonetheless, in order to go from A to B, the double dimers P1-P2 and P3-P4 must be destroyed.This is impossible if the mass m → 0, and the system is frozen in the A state.

A

B

Figure 6.5: An almost saturated configuration of polymers (baryonic loops). Monomers A andB are not neighbours, so in order to eliminate them, the surrounding baryonic loops must berearranged. This is quite unlikely to happen if the mass is very low and the chemical potentialhigh, producing unexpected defects in the lattice.

of high winding number are rarely produced (by the dynamics of the simulation). Nevertheless,even at large masses, at some point one of these large baryonic loops with high winding numbercan be constructed by chance. Although the longer the loop, the higher the probability of beingdestroyed in the next Monte Carlo step, the great weight of loops with high winding numberoutcomes by far the increase of Monte Carlo attacks due to the length of the loop: They aresuperstable and non-interacting objects, which stay in our lattice during the whole simulation.

112

These loops are sometimes twisted in strange ways, wasting a large volume, and preventingthe formation of new loops, even at great values of µ. This twisting truly can spoil simulationresults, for the loops are not allowed to be deformed without being broken and rebuilt.

On the whole, the algorithm proposed by Karsch and Mutter is bound to lead to manyartifacts, which are not related to the polymerization formulation of QCD at strong coupling,but on the particular implementation of this polymerization on the computer. We expect toobtain relable results with this algorithm only for largely massive fermions m ≈ 1. For largemasses, however, the sign problem does not seem to be relevant [120].

Indeed this naive implementation was used by Rossi and Wolff, and Karsch and Mutter;and their results, albeit remarkable, are far from satisfactory. The whole idea of combiningpolymerization of the fermionic action with the clustering of configurations to remove the signproblem is brilliant, but the implementation at that time was poor. This is what we tried tofix.

6.4 The Improved Monomer-Dimer-Polymer (IMDP ) algorithm

It is clear that we need a proposal to modify the configuration which can deal with these bootle-necks. At first, one can think of keeping the dimer ↔ monomer modification and try to performseveral changes around a site before the metropolis accept/reject step. This enhacenment de-creases the locality, and therefore, the acceptance rate, but it should improve the ergodicity. Ithappens that the decrease in the acceptance rate is too large (at some point in our simulations,it would drop to zero) to be considered a feasible solution. The only remaining possibility is toplay with dimers.

The first modification, allows a single monomer to be displaced freely around the lattice byrotating a dimer of an opposite point around (see fig. 6.6). This possibility allows two freshlycreated monomers to be separated, and two distant monomers to be rejoined. The probabilityof this kind of modification to happen is of order O(1) (unless the creation or annihilation offermionic loops is involved).

M

BA

M

Figure 6.6: An example of monomer displacement. The monomer M moves freely around theallowed sites, and after one montecarlo step is in position to annihilate with the other lonelymonomer, removing the point defects of the lattice.

This solves the problem of the isolated monomers.

113

Our second proposal deals with only dimers, and thus it becomes quite handy when dealingwith small, or even vanishing masses. A dimer is allowed to be displaced around any closedpath in this formulation, and we choose the simplest closed path (the plaquette) as proposalto modify configurations, as any other larger closed path can be constructed from plaquettes.Again, the probability of accepting the new configurations generated using this modification isof order O(1), if the baryonic loops do not enter in the calculus. This way, the problem of thechiral limit is solved, and the ergodicity of the algorithm, almost completely restored.

A B

D D

Figure 6.7: An example of dimer rotation around a closed path. The difference between A andB lies in the loop marked as D. Example 6.4 can also be solved without creating monomers intwo steps of dimer rotations.

The last modification deals with the lack of interaction among baryonic loops. It would beinteresting to be able to split or merge neighbouring loops, as the modulus of the weight ofthe configurations with the loops merged/splitted is of the same order of magnitude7, but themodifications proposed up to now require the participant loops to be broken and then rebuilt.Taking into account the high weight of a baryonic loop, this represents a significant bottleneckin the simulation procedure.

Thence, the following merge/splitting options where added. In fact, this modification alsoenables the algorithm to deform existing loops, by combination of spatial loops baryonic loopswith temporal loops. Thanks to this last modification, loops of very high winding number canbe created and destroyed at ease by the algorithm. Indeed we observed non-permanent loopsof large winding numbers (≈ 50) in our simulations.

The probability of all these modifications is (in most cases) of order O(1), that is whythey are so important. At high masses, they compete with the original dimer ↔ monomerstransformation of Karsch an Mutter, but at low masses, they become dominant and rule thedynamics of the system.

IMPD vs Worm Algorithm

The similarities of these modifications with the worm algorithm are, at least, notorious. Itseems that we have implemented some sort of worm algorithm without worms, i.e., without

7If the loops merged are both temporal loops, the sign of the configuration might change after this modification.This enhancement of sign switching, which is necessary for ergodicity, might play against us later.

114

open chains. All the configurations generated by our procedure are physical, opposing whathappens in the worm algorithm. This similarity is, surprinsingly, completely accidental. In fact,we learned on the worm algorithm after developing these modifications. We would probablyhave used the worm algorithm directly to treat the MDP model (as Ph. de Forcrand did in[112, 117]) if we had known about it.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68

Bar

yon

icden

sity

nB

Chemical potential µ

m = 0.000

×××××××××××××××××××××××××××

×××××××××

m = 0.025

+++++++++++++++++++++

+

+++++++++++++

+m = 0.050

⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆

⋆⋆

⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆ ⋆m = 0.100

△△△△△△△△

△

△△△△△△△△△△△△△△△△△△△△△△△△△△ △

Figure 6.8: Comparison of our results to those of Philippe de Forcrand in [112]. Ph. de Forcrandresults are always marked in circles. The agreement is good within errors, being the errorbarssmaller than symbols. To obtain the plotted data, we used 100000 thermalization steps, followedby 100000 measurement steps.

The question of which of the two, the worm algorithm or our approach, bests the other innumerical simulations is not very important. As they are based in similar principles, we expectboth of them to perform almost equally, after doing the corresponding optimizations. And weexpect both of them to share the same problems in the same situations. We have, however, togive the worm algorithm an edge when it comes to computing correlation functions, for the non-physical configurations generated during the update procedure of the worm algorithm becomequite handy for this task.

Numerical results

We performed simulations at several volumes to check whether the sign problem was a realissue on this model or not, mainly in the chiral limit (for high values of the mass, the Karschand Mutter algorithm should work properly). Our simulations were performed starting from acold configuration (system filled up with baryonic loops, nB = 1, and high initital value of thechemical potential µ, that decreases along the simulation), and from a hot configuration (systemempty of loops, nB = 0, and a low value of µ, increasing as the simulation goes on). We left thesystem thermalize for 1000000 iterations, allowing it to relax to a new configuration, and thenperformed measurements in 200000 montecarlo steps. The results for the smaller volume beganto reveal a sign problem after the introduction of our modifications of the algorithm, althoughfor V = 44 only in a small region of µ’s the average sign takes problematic values.

This behaviour contrasts with the disparition of the sign problem for high values of µobserved in the old algorithm. The reason why the sign problem was tractable in the originalwork of Karsch and Mutter is the fact that the system easily saturates with baryonic loops, andthere is only one saturated state, which carries a positive weight. If the system spents most of

115

the time in this state, the mean sign will take high values. In fact, the system usually stays inthe saturation state forever, unless the chemical potential is reduced.

In our simulations, the system did not always succeeded in finding a saturated state, due tothe presence of dislocations and strange geometries for the loops, allowed by the new improve-ments of the algorithm. It stayed in an almost saturated state, where the system evolved quiteslowly, for there was no room in the configurations to propose modifications without breakinga large number of baryonic loops. So at first glance, the improvements in the ergodicity of thealgorithm turn out to increase the severity of the sign problem, or at least, the extension of theregion where it appears.

The saturated (or almost saturated) state seems a finite volume effect, rather than a phys-ical property of QCD [101]. In fact, after increasing the spatial volume we observe how thesaturation effect is relaxed, and the mean sign drops.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.45 0.50 0.55 0.60 0.65 0.70 0.75

Chemical potential µ

Baryonic density nB

××××××××××××××××××××

××××××

×××××××

×××

×××××

××××××××××××××××××

×Mean sign 〈σ〉

¤

¤¤

¤¤

¤¤

¤¤

¤¤

¤

¤

¤

¤

¤¤

¤

¤

¤

¤

¤

¤¤¤

¤

¤¤

¤

¤¤¤

¤

¤¤

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

¤

Figure 6.9: Baryonic density and mean sign for 44 volume at zero mass, hot start. The systemdoes not saturate completely, probably because some long-lived complex loop structures arecreated. This allows a sign flipping, even at very high µ′s. Errors are not shown to improvereadability, but they are smaller than symbols in those regions where the mean sign 〈σ〉 ∼ 1and huge where 〈σ〉 is small or fluctuates strongly.

The explanation of the lowering of the mean sign value as the spatial volume increases is quiteintuitive: As µ increases, baryonic loops are more likely to be created and stay. Nonetheless,these loops have a higher life expectancy if they are packed. This is because, in order todestroy a loop inside a pack of loops, we usually need to mess with the neighbouring loops.Since those loops have quite large weights, proportional to cosh (3LT µ), the greater the numberof loops involved in the modification of the configuration, the harder to destroy them. Inother words, the loops which are close together protect each other. So the loops tend to pack,and the straight loop (which carries a positive weight) is the best loop for packing, achievinga greater loop density. But the loops at the boundaries of the pack are more exposed tomodifications, deformations and annihilation. These boundary loops change easily of shape,and so does the sign associated to them, that keeps flipping during the simulation. Indeed,for very large volumes, the saturation state is difficult to reach, for the loops tend to disorder,merge and arrange in complicated geometrical ways. Dislocations and defects are produced,and the system never saturates. It seems that the loop entropy plays a role here.

The saturation also introduces large autocorrelation times in the simulations, slowing down

116

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

Chemical potential µ

Baryonic density nB

××××××××××

×××××××××××××××××××××××××××××××××××××××××

×Mean sign 〈σ〉

¤¤¤¤¤¤¤

¤¤¤

¤¤ ¤ ¤ ¤¤¤

¤

¤ ¤

¤

¤

¤

¤

¤

¤

¤

¤

¤

¤¤

¤¤¤

¤

¤¤¤

¤

Figure 6.10: Baryonic density and mean sign for 83 × 4 volume at zero mass, hot start. Onlywhen the system saturates completely µ ∼ 0.87, the mean sign stops fluctuating wildly. Errorsare not shown to improve readability, but they are smaller than symbols in those regions wherethe mean sign 〈σ〉 ∼ 1 and huge where the 〈σ〉 is small or fluctuates strongly.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

Chemical potential µ

Baryonic density nB

××× ×××××××××××××××××××××××××××××××××

×

×××××

××

×

×

××××

×Mean sign 〈σ〉

¤¤¤¤¤¤¤¤¤¤¤¤

¤

¤¤¤¤ ¤¤

¤

¤

¤

¤

¤

¤

¤

¤

¤

¤

¤¤¤

¤

¤

¤ ¤ ¤¤

¤

¤

¤

Figure 6.11: Baryonic density and mean sign for 84 volume at zero mass, hot start. The systemdoes not saturate at the µ values simulated, and the mean sign fluctuates without control,mostly after the critical chemical potential µ ∼ 0.8 has been reached. Errors are not shown toimprove readability, but they are smaller than symbols in those regions where 〈σ〉 ∼ 1 and hugewhere 〈σ〉 is small or fluctuates strongly.

the measurements. Almost saturated states show great stability, leading to highly correlatedmeasurements. This is a point where one might expect the worm algorithm to perform betterthan ours, but the advantage is not very clear, for each time the worm algorithm creates anew valid configuration, non-local modifications are introduced, and when the system is in analmost saturated state, these modifications are bound to involve many loops, and therefore, tobe discarded8.

8The low autocorrelation times shown in [112] happen at very high temperatures LT = 2, where the barynonic

117

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.45 0.50 0.55 0.60 0.65 0.70 0.75

Bar

yon

icden

sity

nB

Chemical potential µ

Hot Start

××××××××××××××××××××××××××××××××××××××××××××××××

×××

××

×

×××××

×Cold Start

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

¤

Figure 6.12: Hysteresis cycle for 64 at zero quark mass. The cycle is greatly reduced withrespect to [111]. The errorbars are not plotted to improve the readability of the figure. Theseare smaller than symbols in the saturated (nB = 1) and empty (no loops, nB ∼ 0) states, andincredibly large in the region in between. The simulation starting from a hot configurationnever reaches the saturation state due to the presence of dislocations and complex structures ofloops.

Another feature of this model is the hystheresis cycle it presents. The hysteresis cycle wasfirst observed and discussed in [111]. In our improved algorithm, it is greatly reduced, butit exists as well. This large hysteresis seems related to the saturation of loops as well: Thesaturation state is superstable, and any modification aimed to destroy it must either create apair of monomers, which is a quite improbable process for low masses, or destroy several loopsat the same time.

This two features, (i.) a superstable saturated state produced by a finite spatial volumeeffect, and (ii.) the presence of a severe sign problem in the region close to the critical valueof µ, makes us conclude that this model does not solve in any way the sign problem for QCDat finite density. However, all these effects are relaxed for very high temperatures, where theweight of the loops is greatly reduced, and the model works well, as fig. 6.8 shows. This is nota great deal, and there are other procedures which also work at high temperatures for any valueof β [121].

6.5 Conclusions

The MDP model developed by Karsch and Mutter was quite successful at solving the signproblem in a polymer formulation of QCD at µ = 0 and for non-zero masses. As this systemseemed quite manageable, the extension to µ 6= 0 was mandatory; unluckily the original MDPimplementation suffered from ergodicity problems at small values of the mass, which only allowthe system to live in two distinct states: Either complete saturation or complete depletion ofbaryonic loops. Each one of those states does not suffer from the sign problem, this is quiteevident in the depleted configuration, and almost trivial in the saturated one: In order tosaturate the lattice, only the simplest loops (the straight ones) are allowed. These loops are the

loops are easy to break, and our algorithm is also expected to perform well.

118

most important contribution to the partition function, and carry positive weights. Thence, nosign problem seems to appear in the MDP model.

Nonetheless, once the ergodicity problem has been solved by improving the simulation al-gorithm, this picture changes. Now there are more allowed states, and these states need not bepositive-weighted. In order to see this effects we need large volumes, for the system sees thewalls9 of our box very soon. This last point ‘solves’ the sign problem for small volumes and lowtemperatures10, because it leads to saturation of baryonic loops, but this is an unrealistic effectwhich leads to unrealistic results [101, 102, 103]. If the volume becomes larger, we see a bunchof baryonic loops in the center of our box, but the borders are empty, and the system is notsaturated, but free to evolve. As the loops can move and change its shape at the borders, andat large µ the loops carrying negative weight are as important as the ones which carry positiveweight, both kind of loops are expected to be created and destroyed at the border of the box,leading to a severe sign problem. If, on the contrary, the volume is too small, the system satu-rates easily. As in total saturation, only positive loops are allowed, the sign problem dissapears.But this is rather a finite volume effect.

Every advocate of this model wields the same argument to defend the model, and to explainwhy it works: The positive-weighted loops are more important than the negative counterpartsin the partition function, so eventually, the positive-weighted loops will take over, and the signproblem will disappear. This argument is quite naive. The larger weight of the positive-weightedloops might not be enough to overcome the sign problem. In fact, the positive-weighted loopsmust have weights much larger than those of the negative-weighted loops in the thermodynamiclimit if we want to see a mean sign different from zero, and for sure this is not the case of theMDP model as the temperature T → 0 while µ 6= 0. This fact can be checked in a very simplemodel featuring a severe sign problem.

Imagine a linear chain of non-interacting spins. Each spin can point either upwards ordownwards, as in the Ising model. If a spin points upwards, it contributes with a factor p > 0to the configuration weight, and if it points downwards, it contributes with a factor q < 0. Letp − q = 1 to normalize those probabilities, and let us try to compute the mean sign 〈σ〉 of thismodel for a given volume V :

〈σ〉 =

(p + q

p − q

)V

= (p + q)V .

As p+ q < 1 always, the mean sign (and Z itself) exponentially vanishes in the thermodynamiclimit. It does not matter how greater is p against |q| (take for instance values p = 0.99 andq = −0.01), the sign problem is severe, unless, in an effective way

limV →∞

∣∣∣∣

q

p

∣∣∣∣= 0

For this to happen, the ratio q/p must become small as the volume increases. Imagine that

p = 1 − κV −α q = −κV −α,

being α an arbitrary number higher than zero. Then the mean sign would become

9For the (anti-)periodic boundary conditions used in this work, there are no walls, but nonetheless the systemsees itself through the boundary, and finite volume effects become important.

10At high temperatures (small LT ) the MDP model work well in general, giving high values for the mean sign.The temperature can be fine tuned using a parameter, introduced in the original paper of Karsch and Mutter[109]. As we wanted to perform simulations at zero temperature, we have not included this parameter in thiswork, although Ph. de Forcrand made use of it [112, 117].

119

〈σ〉 =(1 − 2κV −α

)V=

(

1 +1

−V α

2κ

)−V α

2κ

−2κV 1−α

= e−2κV 1−α

.

For α > 1, 〈σ〉 → 1 in the thermodynamic limit, and there is no sign problem at all. For α = 1the sign problem can be handled in the infinite volume limit, as 〈σ〉 → tends to a fixed value.If 1 > α > 0, the sign problem is almost exponentially severe, and can be treated by usingthe theory of the modulus, i.e., taking all the weights as positive and hoping that the resultingtheory is equivalent to the original one in the thermodynamic limit; and this is what happensindeed, although the convergence might be slow. Only in the case α = 0 the mean sign goes tozero exponentially with the volume. In the case of the MDP model, the loops carrying negativeweights are almost as simple as the positive ones, and have weights which are almost equallylarge. If the configurations carrying positive weights took over the negative ones, the theoryof the modulus would be equivalent to the original theory. But we know for sure that this isnot the case: the negative weights in the configurations come from the loop expansion of thefermionic determinant, and are necessary in order to reproduce correctly the anticommutationrules. As we expect at very high densities the quark dynamics to be ruled mostly by Fermi-Diracstatistics, it is quite natural to find a severe sign problem in this formulation. In the µ = 0case, the model has the same problem, but the fermionic cancelations are introduced by handexplicitly, thence it can be simulated. If we removed these cancellations, and took the modulusof the resulting theory (dropping the sign on the negative weighted configurations), then theresulting theory would feature hard-core bosons11 instead of fermions. Therefore, the theory ofthe modulus can not be equivalent to the original one. On the whole. things seem to be morecomplicated than what the advocates of this model defend.

The failure of the MDP model, showing a severe sign problem for µ 6= 0, is a heavy blowto the finite density QCD practitioners. However, there are other possibilities, some of themopen only to discrete systems, that we have not explored: Merom algorithms, a refinement ofthe clustering techniques, or the addition of other terms to the action which, in combinationwith clustering, might help reduce the sign problem. That is why we should keep investigatingthese models.

11A hard-core boson is a field φ with Bose statistics, but verifying a nilpotency rule φ2 = 0.

120

Chapter 7

Algorithms for the pure gauge action

“Calculation never made a hero.”

—John Henry Newman

Although the MDP −QCD does not work very well for µ 6= 0, it can be taken as an alterna-tive for fermion simulations at zero chemical potential. Since the MDP model escapes from thecomputation of the determinant, it should represent a great algorithmic advance for simulatingfermions on a lattice. Moreover, for non-zero µ, there are still some unexplored possibilities (likemixing merom cluster algorithms with this formulation) that can reduce the severity of the signproblem. Nevertheless, the model has an important drawback which constrains its range ofapplication: It is only valid at strong coupling. As the continuum limit of QCD lies in the weakcoupling limit, this restriction can not be regarded slightly, and must be faced seriously, if weaspire to make from the MDP approach a competitive model for QCD. In other words, weneed to treat the gauge action.

One way to confront the problem is to mimic the procedure applied to fermions, i.e., (i.) theexpansion of the fermionic action and (ii.) the integration of the Grassman variables to yield aset of graphs, and try to apply this method to the pure gauge action. We can expand the gaugeexponential in β powers, and integrate the link variables of each term of the expansion.

In fact, this procedure defines a general transformation of the terms of the partition functionwhich can be applied to any model. This transformation maps a continuous system into adiscrete one, usually with a sign problem, except for some particular cases. Nonetheless, thereare several techniques, ready to be applied to discrete systems, which might reduce or evensolve completely the sign problem, like configuration clustering, or the merom algorithm, yetthe integration of the latter with polymeric models remains unexplored.

Another advantage of the simulations of these discrete systems –when compared to theirstandard counterparts- is the fact that, when it comes to fermions, they become fast. In theformer section, devoted to the MDP model of Karsch and Mutter, we transformed a ratherheavy computational object, the fermionic determinant, into a set of balls and sticks (monomersand dimers), and as we will see, the graphs do not get much more complicated when we introducegauge fields.

Thence, although the polymeric models at this point are not completely developed, forthey must face a sign problem in order to be useful for the lattice community, there are manypossibilities in the horizon to fix this caveat. It is reasonable to start the research on the gaugepart of the action, and try to reconcile it with the fermionic part.

121

7.1 The strong coupling expansion for Abeliangroups

To begin with, we consider only abelian groups. Of course, we are interested in applying thisideas to SU(3) in the end, but in order to develop a systematic method to construct the graphs,we decided to start from the easiest models we could. These are the U(1) and Zn gauge fieldtheories on the lattice, formulated with the usual Wilson action

SPG = −Re

[

β∑

n,µ,νµ<ν

U(n)µU(n + µ)νU†(n + ν)µU(n)†ν

]

=

−β

2

NP∑

k=1

(

Uk + U †k

)

, (7.1)

where k indexes the NP plaquettes of the lattice, and Uk and U †k are the oriented product of

gauge fields around the plaquette k, and its complex conjugated, which I will call anti-plaquette.The partition function can be expanded in powers of the inverse of the coupling β

Z =

∫

[DU ]

NP∏

k=1

eβ2

“

Uk+U†k

”

=

∫

[DU ]

NP∏

k=1

∞∑

n=0

(β2

)n

n!

(

Uk + U †k

)

=

∫

[DU ]

NP∏

k=1

∞∑

n=0

(β2

)n

n!

[(n

j

)

(Uk)j(

U †k

)n−j]

=

∫

[DU ]

NP∏

k=1

∞∑

j1,j2=0

(β2

)j1+j2

(j1 + j2)!

[(j1 + j2

j

)

(Uk)j1

(

U †k

)j2]

. (7.2)

The next step consist on the expansion of the product over the plaquettes, to find a sumover products of different powers of the plaquettes, multiplied by constant factors. After thisexpansion has been done, the integration of each summand of plaquettes can be carried outjust by following the group integration rules: The integral of every non-trivial product of linksis zero. In other words, the only surviving terms are those where each plaquette Uk is, eitherannihilated by its corresponding anti-plaquette U †

k , or belongs to a close surface of orientedplaquettes1. In the case of the Zn groups, the n-th power of a (anti-)plaquette is non-trivialas well2. Thus, we can label each term of the expansion with two numbers (nα

k , nαk ), being α a

labels referring to a specific term in the expansion, k the label for the plaquette on the lattice,and consequently, nα

k (or nαk ) the power of the k plaquette UP (or anti-plaquette U †

P ) in theterm α of the product expansion. After integration, the partition function is a sum of differentterms indexed by α,

Z =∑

α∈CCα (β) , (7.3)

1By oriented I mean that half of the faces are plaquettes, and the other half anti-plaquettes, placed in a certainway to ensure that every link is annihilated.

2This non-triviality of the n-th power of a link was the origin of baryons in the MDP model. We expect anyZn gauge theory to feature baryons composed of n quarks as well.

122

with C standing for the space of graphs (the different summand terms in the partition function).The particular value of Cα (β) is given by the group integration of the term α of the expansion,and its result can be computed analytically

Cα (β) =∏

k

βnαk +nα

k

2nαk+nα

k nαk !nα

k !. (7.4)

The key point underlying this expansion (and the MDP one) is the interpretation of theseCα as configurations of a new system, equivalent to the original one. Each configuration α isgiven by the set of numbers (nα

k , nαk ), and its probability is given by wα = Cα/Z. Montecarlo

simulations can be performed in this configuration space, instead of using the original one.

The configurations are translated in our lattice into a set of closed surfaces of plaquettesand anti-plaquettes. The simplest closed surface that we can imagine on the lattice is the cube,and we can regard any other closed surfaces as a particular spatial arrangement of cubes. Thisis one of the most beautiful properties of this algorithm: The fact that it allows a geometricalinterpretation. We can think of a lattice where there are open surfaces made of plaquettes andantiplaquettes, closed surfaces (volumes) and other more complex structures, evolving as timegoes by. To create, destroy and modify these surfaces in all the imaginable ways, we only needtwo transformations acting as building blocks for the U(1) group, and three for the Zn groups3:

(i.) The transformation (nαk , nα

k ) → (nαk + 1, nα

k + 1), and its inverse (when nαk , nα

k > 0)(nα

k , nαk ) → (nα

k − 1, nαk − 1). This transformation creates a pair plaquette/anti-plaquette

at a given place k of the lattice.

(ii.) The addition of a closed cube of oriented plaquettes, for dimensions higher than two. Anyclosed surface can be constructed by adding cubes and removing the touching faces usingthe first transformation.

(iii.) In the case of a Zn group, the addition of n plaquettes or anti-plaquettes to the sameplace k, that is, (nα

k , nαk ) → (nα

k + n, nαk ) or (nα

k , nαk ) → (nα

k , nαk + n). This transformation

keeps the rule nαk = nα

k (mod n).

A Monte Carlo designed with these transformations for the configurations can generate almostany possible configuration; the only exception are the plaquette foils, which will be treatedlater. The efficiency of our particular implementation will be compared against the standardheat-bath algorithm.

We must be careful with the implementation of cubes. In three dimensions, there is only onepossible cube per lattice site4, and the orientation of the plaquettes does not matter, providingthe gauge links are annihilated as the Bianchi identity dictates (if there is a plaquette in oneface, there must be an anti-plaquette in the opposing face). But in four dimensions, thereare four possible cubes per lattice site. The orientation of the plaquettes of the first cube canbe chosen arbitrarily, as long as we comply with the Bianchi identity (in which face there isa plaquette, and in which face there is an antiplaquette), but this imposes constraints in theorientation of the other three cubes, for they share common faces. All the cubes must haveconsistent orientations.

3The addition or removal (when possible) of a single surface of plaquettes or anti-plaquettes which wrapsaround the lattice should be an allowed transformation as well; however, this is a finite volume correction, andwe decided to ommit it in the simulation program.

4Assuming a cubic lattice.

123

Figure 7.1: Fixing the orientations of the plaquettes for a single cube. Once a face is locked inorientation, its opposite face must be inversely oriented.

7.2 Measuring observables

The computation of observables is quite easy in this representation. The definition of theobservable plaquette5 is

〈P¤〉 =1

NP∂β ln (Z) (7.5)

Using (7.3) and (7.4), we obtain

〈P¤〉 =1

NP∂β ln

(∑

α∈CCα

)

=1

NP

1

Z

∑

α∈C

(nα + nα)

βCα (7.6)

with

nα =

NP∑

k=1

nαk nα =

NP∑

k=1

nαk

The quantity Cα/Z is the probability wα of each configuration α. The mean of the observableplaquette is then

〈P¤〉 =1

βNP

∑

α∈Cwα (nα + nα) (7.7)

that is to say, the observable plaquette is equal to the mean value of the sum of the occupationnumbers nα plus nα, divided by β and normalized by Np.

Another interesting observable is the specific heat:

CV = ∂β〈P¤〉 (7.8)

We can profit from the previous expression of P¤ (7.7) to find the following equation

5We must remark that the observable plaquette and the plaquettes living in our lattice are not the same,although they are strongly related. The observable plaquette refers to the minimal Wilson loop, whereas theplaquettes refers to geometric entities, living on the lattice. In order to keep the discussion clear, we will alwaysrefer to the minimal Wilson loop as observable plaquette.

124

CV =1

NP

∑

α∈Cwα (nα + nα)2

β2−

(∑

α∈Cwα (nα + nα)

β

)2

−

∑

α∈Cwα (nα + nα)

β2

}

= NP

[⟨P 2

¤

⟩− 〈P¤〉2 −

N2P

β〈P¤〉

]

(7.9)

Sometimes, it is interesting to compute correlation observables, such as the Wilson loop(larger than the single observable plaquette), or the plaquette–plaquette correlation function.To compute these it will prove helpful to introduce a pair of variable coupling constants {βk, βk},which depend on the plaquette site k, in such a way that the partition function reads now

Z(βj , βj) =

∫

[dU ]∏

k

e

“

βk2

Uk+βk2

U⋆k

”

(7.10)

The weight of the configurations changes accordingly

Cα(βj , βj) =

NP∏

k=1

βnα

k

k βnα

k

k

2nαk+nα

k

(nα

k !) (

nαk !

) (7.11)

Now the correlation functions or the Wilson loops are computed by simple derivation, andthen taking all the βk, βk to the same value. For instance, the 2 × 1 Wilson loop can becalculated as

〈PW2×1〉 = 22 lim

βj ,βj→β

∑

α∈C ∂βk∂βk+1

Cα(βj , βj

)

∑

α∈C Cα(βj , βj

) =

22∑

α∈C

nαknα

k+1

β2wα (7.12)

where k and k + 1 are contiguous plaquette sites. The generalization of this result to largerplanar Wilson loops is straightforward. Indeed, the expectation value of any planar Wilson loopcan be computed as the mean value of the product of the occupation numbers of the plaquettesenclosing the loop, multiplied by a factor 2/β to a power which is the number of plaquettesinvolved. This observable is quite remarkable, for it is computed as a product of occupationnumbers, and features a (2/β)Area factor, which eventually may become exponentially large orsmall as the size of the loop increases. All these particular facts render this observable hard tocompute, as we will see in the numerical results.

In the same way we can also obtain the correlation functions for two arbitrary plaquetteson the lattice:

〈UkUl〉 = 22 limβj ,βj→β

∑

α∈C ∂βk∂βl

Cα(βj , βj

)

∑

α∈C Cα(βj , βj

) = 22∑

α∈C

nαknα

l

β2wα (7.13)

and

〈UkU∗l 〉 = 22 lim

βj ,βj→β

∑

α∈C ∂βk∂βl

Cα(βj , βj

)

∑

α∈C Cα(βj , βj

) = 22∑

α∈C

nαk nα

l

β2wα (7.14)

These expressions are quite analogous to the formulae derived for the Wilson loop.

125

Finally from (7.13) and (7.14), and taking into account the symmetry of the model, we canwrite:

〈ReUkReUl〉c = 1β2 〈(nk + nk) (nl + nl)〉 − 〈nk + nk〉 〈nl + nl〉 (7.15)

〈ImUkImUl〉c = − 1β2 〈(nk − nk) (nl − nl)〉 (7.16)

where the brackets denote average over configurations.

Plaquette foils and topology

There is an allowed object we have not make used of in our expansion: A foil of (anti-)plaquettes,forming a surface, which wraps around itself through the boundary conditions. It is an objectcontaining roughly L2 plaquettes, with L the length of our lattice, and thence, the probabilityof creating such an object is negligible for β < 2 as V → ∞. For higher values of β however, thisobjects become important. The problem is the fact htat we can not generate such a plaquettesheet using cubes and standard plaquettes. We can only create pairs of sheets6. We mightwhink at first that this difference might give some contribution, but as we expect to createthe same number of foils and anti-foils7, the contribution should cancel out. If we have chosento remove the sheets from our simulation programs is because they lead to important finitevolume effects, related to those happening in the heat-bath [122]: if a sheet is created during asimulation, and it is deformed by the addition of plaquettes and cubes, it might be quite hardto remove, slowing down the simulation critically. As the contribution of these foils should notbe important (they take the system to a local minimum), it is better to left them behind.

Numerical work

In order to see our algorithm (which we should call, from now on, geometric algorithm) at work,we have performed numerical simulations of several lattice gauge theory systems in three andfour dimensions. Our aim is to check the goodness of our approach, comparing the results weobtain using the geometric algorithm with those coming from more standard ones; hence wewant to compare the properties of the algorithm itself, in terms of numerical efficiency andautocorrelation times with, for example, the usual heat-bath algorithm. We have in mind theresults of [123], where it was claimed that, with a similar algorithm, at a second order criticalpoint in a non-gauge system, there is a very strong reduction in the critical slowing down.

Let us start with the three dimensional U(1) lattice gauge model: this model is known tohave a single phase from strong to weak coupling. We have chosen to measure two simpleobservables, namely the plaquette observable and the specific heat, following the definitionsgiven in the preceeding Section. We have simulated the model with our algorithm and with astandard heat-bath for a large interval of β values using a 123 lattice; we allowed the system toreach thermalization for 5×104 iterations and then measured the observables for 106 iterations.Errors were evaluated using a Jackknife procedure. The results are shown in Fig 7.2.

We can easily see in this figure that the two simulations give essentially the same results.

Almost the same situation can be depicted also for the four dimensional U(1) model; theresults of a similar set of simulations, performed with the two algorithms on a 164 lattice, areshown in Fig 7.3.

Here the only difference can be seen near the phase transition point. Remember that dueto the difference in finite volume terms between the two algorithms, the precise pseudo-critical

6For instance, filling a plane with cubes and removing the edges at the boundaries of the lattice would resultin a pair foil/anti-foil.

7We do not expect a breaking of the symmetry plaquette/anti-plaquette.

126

0 0

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

1.0 1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

Av

erag

e P

laq

uet

te

Sp

ecif

ic H

eat

Beta

Heat-BathGeometric Algorithm

Heat-Bath CVGeometric Algorithm CV

Figure 7.2: Three dimensional U(1) lattice gauge system. Errors are smaller than symbols.

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0

Aver

age

Pla

quet

te

Spec

ific

Hea

t

Beta

Heat-BathGeometric Algorithm

Heat-Bath CVGeometric Algorith CV

Figure 7.3: Four dimensional U(1) lattice gauge system; please note the different scale for thespecific heat on the right. Largest errors (those on the pseudo-critical point) are smaller thansymbols.

coupling value at finite volume has to be slightly different. We have calculated a few morepoints very close to the critical beta for each algorithm, and the results can be seen in tables 7.2and 7.2. It seems that the peaks are sharper (narrower and larger) in the heath-bath algorithm(take notice of the much larger value of the specific heat for some of these points as comparedwith fig.7.3).

These results allow us to infer that the geometric approach is able to reproduce all thefeatures of the models under investigation, and when differences are seen, they can be easily ex-plained on the difference between finite volume terms. To study more carefully these differences

127

β Plaquette Cv

1.010700 0.62545(34) 24.1(3)1.010850 0.64014(143) 88.2(2)1.010900 0.64940(54) 47.9(4)1.011100 0.65473(6) 2.6034(1)1.011600 0.65576(12) 2.5559(2)

Table 7.1: Heat-bath results near βc.

β Plaquette Cv

1.011120 0.6373(45) 67.2(7)1.011160 0.6423(46) 67.4(3.6)1.011420 0.6549(1) 2.9660(2)

Table 7.2: Geometric results near βc.

we have calculated βc(L), the critical coupling for each algorithm at different lattice sizes. Wepresent in table 7.3 the results. We also include in the table the results of a fit of βc(L) for thethree largest lattices from each set to the finite-size scaling law expected for a first-order phasetransition [124], in order to obtain the value βc(L = ∞). This gives a good fit and consistentresults for the infinite volume limit.

L βc(L) (heat-bath) βc(L) (geometric)

6 1.00171(8) 1.00878(20)10 1.00936(11) 1.01062(7)12 1.01027(8) 1.01100(7)14 1.01064(4) 1.01103(20)16 1.01084(17) 1.01116(14)∞ 1.01108(10) 1.01120(22)

Table 7.3: βc(L) for heat-bath and geometric algorithm respectively.

The presence of two clearly different phases in this model, namely a confining and a Coulombone, allows us to study the behaviour of the Wilson loop results in two different physicalsituations; as above, we have also performed standard simulations for a cross check between thetwo approaches. In Figs. 7.4 and 7.5 we report the behaviour of the Wilson loop in both phases(confining in Fig. 7.4 and Coulomb in Fig. 7.5) and in lattices of different size (124, 144, 164).

These figures deserve some comments. The geometric algorithm seems to suffer from largerstatistical errors than the heat-bath method, regardless of the phase of the system. To un-derstand this result, we should have a close look at the inner machinery of the algorithm, inparticular, the way the Wilson loop is computed (see eq. (7.12)). First of all, the mean valueof the Wilson loop is computed as a sum of integer products, implying the existence of largefluctuations between configurations. For example, doubling the occupation number of a singleplaquette doubles the value of the loop. This is a quite common fluctuation at the β valuesof our simulations, and the fluctuations will increase as the loop (and therefore the numberof plaquettes) grows. To complicate the computation further, we are trying to calculate anexponentially small quantity by summing integer numbers. The discrete nature of this compu-tation tells us that non-zero values of the quantity must appear with an exponentially smallprobability. This explains the inherent difficulties of the large Wilson loops (4× 4 and greater)measurement in the confining phase. The result is shown in Fig. 7.4: the mean value of the5×5 Wilson loop was exactly zero in the geometric algorithm, which is of course wrong. Finally,the expectation value of the Wilson loop is proportional to a (2/β)A factor, with A the looparea. This value may become huge (or tiny) for large loops and low (or high) values of beta,thus enhancing the problems that arise from the discreteness of the algorithm 8.

8See [125, 126] for a discussion of some numerically efficient algorithms for the calculation of large Wilson

128

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 5 10 15 20 25

Lo

op

Val

ue

Loop Area

Heat-BathGeometric Algorithm

Figure 7.4: Real part of the Wilson loop versus the loop area for the confining phase (β = 0.9)in the four-dimensional U(1) gauge model. Notice the absence of the 5×5 loop in the geometricalgorithm. The lattice volume was 164.

0.01

0.1

1

0 5 10 15 20 25

Lo

op

Val

ue

Loop Perimeter

Heat-BathGeometric Algorithm

Figure 7.5: Real part of Wilson loop versus the loop perimeter for the Coulomb phase (β = 1.1)in the four-dimensional U(1) gauge model. The lattice volume was 164.

Notwithstanding the stronger fluctuations in the large Wilson loops within the geometricalgorithm discussed above, it has a clear advantage against heat-bath: it does not suffer fromergodicity problems. Indeed the results for the Wilson loop at β = 3 reported in Fig. 7.6strongly support the previous statement. The points obtained with the geometric algorithm

loops and Polyakov loop correlators.

129

0.1

1

0 5 10 15 20 25

Lo

op

Val

ue

Loop Perimeter

Heat-Bath Hot StartHeat-Bath Cold StartGeometric Algorithm

Weak Coupling Approximation

Figure 7.6: Real part of Wilson loop versus the loop perimeter for a large β value (β = 3.0) inthe four-dimensional U(1) gauge model. Notice the difference in performance between the hotand the cold starts of the heat-bath algorithm. The lattice volume was 124.

nicely follow the weak coupling prediction of [127], whereas the heat-bath results for large Wilsonloops, obtained from a hot start, clearly deviate from the analytical weak coupling prediction.The origin of this anomalous behavior in the heat-bath case is related to the formation ofvortices, which are metastable states, that become extremely long lived in the Coulomb phase[122]. These vortices seem to be related to the topological sheets of plaquettes of our model.As we removed them in our simulation algorithm, the GA is expected to perform better thanthe heat-bath.

We have also calculated the plaquette–plaquette correlation (of the real (7.15) and of theimaginary (7.16) parts) in both phases and for plaquettes living in the same plane. Here weexpect a much milder behaviour for the geometrical algorithm, for there is no large (2/β)A

factor, and the fluctuations are reduced to a couple of plaquettes. The results are shown inFigs. 7.7, 7.8, 7.9 and 7.10.

In all the cases, the numerical results obtained with the geometric and heat-bath algorithmsessentially agree, except for the correlations of the imaginary part of the plaquettes in theCoulomb phase (Fig. 7.10), where a clear discrepancy for distances larger or equal than 4is observed. Again in this case the reason for this discrepancy is related to the formationof extremely long-lived metastable states [122] in the heat-bath simulations, which seem to beabsent in the geometric algorithm. Indeed we have verified, with simulations in 124 lattices, thatwhen we start the heat-bath runs from a cold configuration, the disagreement on the correlationsof the imaginary part of the plaquettes in Coulomb phase at large distances basically disappear.There are still small discrepancies in this case, but they can be reasonably attributed to thedifference in finite volume terms between the two algorithms.

To compare computational costs we define a figure of merit, which is the product of thesquared error times the cpu time. We expect the error to vanish like 1/

√NMonte Carlo, and

therefore the quantity defined above should tend asymptotically to a constant. We show thevalue of this quantity for several observables in both phases and for both algorithms in Fig.7.11.

130

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 1 2 3 4

Rea

l P

lq-P

lq C

orr

elat

ion

Distance

Heat-BathGeometric Algorithm

Figure 7.7: Correlation function of the real part of the plaquette versus plaquette–plaquettedistance in lattice units, for the four-dimensional U(1) lattice gauge model in the confiningphase (β = 0.9). Beyond distance 4, the error became far larger than the expectation value ofthe correlation. The lattice volume was 164.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 1 2 3 4

Rea

l P

lq-P

lq C

orr

elat

ion

Distance

Heat-BathGeometric Algorithm

Figure 7.8: Correlation function of the real part of the plaquette versus plaquette–plaquettedistance in lattice units, for the four-dimensional U(1) lattice gauge model in the Coulombphase (β = 1.1). Beyond distance 4, the error became far larger than the expectation value ofthe correlation. The lattice volume was 164.

We can see that the performance of both algorithms is quite comparable. The differencesthat are seen could conceivably change if one were to optimize the specific implementations,but none is obviously much more efficient than the other for the models studied.

131

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 1 2 3 4

Imag

inar

y P

lq-P

lq C

orr

elat

ion

Distance

Heat-BathGeometric Algorithm

Figure 7.9: Correlation function of the imaginary part of the plaquette versus plaquette–plaquette distance in lattice units, for the four-dimensional U(1) lattice gauge model in theconfining phase (β = 0.9). Beyond distance 4, the error became far larger than the expectationvalue of the correlation. The lattice volume was 164.

1e-05

0.0001

0.001

0.01

0.1

0 1 2 3 4 5 6 7 8 9

Imag

inar

y P

lq-P

lq C

orr

elat

ion

Distance

Heat-BathGeometric Algorithm

Figure 7.10: Correlation function of the imaginary part of the plaquette versus plaquette–plaquette distance in lattice units, for the four-dimensional U(1) lattice gauge model in theCoulomb phase (β = 1.1). Notice the different behaviour of the algorithms at large distances.The lattice volume was 164.

In particular, for the plaquette observable and the specific heat, both algorithms have asimilar figure of merit. From our point of view, the differences are not quite significant, and couldchange with careful optimizations. The real plaquette-plaquette correlation is quite another

132

0.01

0.1

1

10

100

Plaquette CV Re CorrR=2 Im CorrR=2

Err

or2

x C

PU

Tim

e G

eom

etri

c/H

eat-

Bat

h R

atio

Observable

Beta 1.1Beta 0.9

Figure 7.11: Ratios of figures of merit for different observables between geometric and heat-bath algorithms. Re CorrR=2 and Im CorrR=2 stand for Real and Imaginary plaquette–plaquettecorrelations at distance 2.

story, for the differences become significative in the Coulomb phase (a factor ≈ 20), but theydo not become worse as β increases, as we test in a 124 simulation at β = 3.0.

On the other hand, the geometric algorithm seems to perform much better for the imaginaryplaquette-plaquette correlation in the confining phase, whereas in the Coulomb phase all theadvantage vanishes. Again, our 124 computation at β = 3.0 reveals that the ratio slowlydecreases as β increases (being ≈ 0.8 at β = 3.0).

Of course, this analysis assumes that both algorithms have no ergodicity problems. Wemust be careful to start from a cold configuration when running the heat-bath simulations inthe Coulomb phase, in order to avoid metastable states which could spoil the reliability of thesimulation.

7.3 The three dimensional Ising gauge model

Let us finally come to the point of critical slowing down: This is a major issue, as any improve-ment in this field can be of paramount importance in term of the cost of large scale simulationsof (statistical) systems at a critical point. Beating critical slowing down is one of the mainmotivations in the development of new Monte Carlo algorithms.

Typically what is found in Monte Carlo simulations of system both in statistical physicsand gauge theories is that the autocorrelationtime τ diverges as we approach a critical point,usually as a power of the spatial correlation length: τ ∼ ξz, where ξ is the correlation lengthand z is a dynamical critical exponent. The typical local algorithm has a value of z ∼> 2,making it very inefficient to simulate close to the critical point. For spin systems there arewell known cluster algorithms with much smaller z. Previously published results [123] on analgorithm similar to ours, but applied to a non-gauge model, have claimed a similarly smallervalue for z. Having also this motivation in mind, we have investigated the autocorrelationproperties of our numerical scheme on the critical point of a system that undergoes a secondorder phase transition (with diverging correlation length). Our model of choice has been the

133

three dimensional Ising-gauge model. We have performed extensive simulations in the criticalzone of this model for several values of the lattice size (and hence correlation length), using boththe geometric algorithm and the standard Monte Carlo approach, the latter known to have alower bound for the autocorrelation exponent z equal to 2, a value typical of all local algorithms.For lattices up to L = 24 we have in all cases more than 5× 105 Monte-Carlo iterations, whichincrease to more than 1× 106 for L = 32, 48, and to more than 4× 106 iterations for the largestlattice L = 64.

For an observable O we define the autocorrelation function ρ(t) as

ρ(t) =〈(O(i) − OA) (O(i + t) − OB)〉

√

σ2Aσ2

B

(7.17)

where the mean values are defined as OA = 〈O(i)〉, OB = 〈O(i + t)〉, and the variances σ2A =

⟨

(O(i) − OA)2⟩

, σ2B =

⟨

(O(i + t) − OB)2⟩

, denoting 〈〉 average over i. We then define the

integrated autocorrelation time by

τ = ρ(0) + 2N∑

t=1

ρ(t)N − t

N(7.18)

where N is fixed, but with N < 3τ and N < 10% of the total sample. In Fig. 7.12 we report theresults for the integrated autocorrelation time of the plaquette versus lattice size in logarithmicscale for both algorithms.

100

1000

20 30 40 50 60 70

Auto

corr

elat

ion t

ime

τ

Lattice length

Figure 7.12: Autocorrelation times at the critical point (of each algorithm) versus lattice length;boxes stand for standard algorithm results, with a linear fit to guide the eye, while circlesrepresent the results of the geometric algorithm. The errors were obtained by a jack-knifeprocedure.

The results of our simulations hint to a different asymptotic behaviour of the autocorrelationtime, although with our present data we cannot obtain a conclusive result. The points for theheat-bath algorithm seem to fall nicely on a straight line, which would correspond to a simple

134

exponential dependence of τ on L, with z = 2.67 ± 0.08, but the geometric algorithm presentsa more complicated behaviour, as well as larger errors. There are signs that the asymptoticbehaviour might be better than for the heat-bath, but much more extensive simulations, outsidethe scope of this work, would be needed to get a definite value for z.

7.4 Conclusions and Outlook

Inspired by the sign problem, we have developed a geometric algorithm, based on the strongcoupling expansion of the partition function, which can be applied to abelian pure gauge mod-els9. We have checked in the U(1) model in 3 and 4 dimensions that the algorithm can beimplemented efficiently, and is comparable with a standard heat-bath algorithm for those mod-els. It seems however that the geometric algorithm does not suffer lack of ergodicity due to thepresence of vortices, as can be the case for heat-bath, depending on the starting point.

We have also studied the algorithm in the 3 dimensional Ising gauge model at the criticalpoint, where we have seen hints that the asymptotic behaviour of the geometric algorithm maybe better than standard heat-bath. This would be very interesting, because in contrast to spinsystems, where there exists cluster algorithms that can greatly reduce critical slowing-down, toour knowledge no similar algorithm is known for gauge systems. Our results are however notenough to establish this, and much more extensive simulations should be done to clarify thispoint.

This algorithmic advance must not be regarded as an isolated contribution, but more pre-cisely as the first step of a more ambitious program, which aims to simulate QCD at finitechemical potential, but which also could be applied to other systems suffering from the signproblem, as for example systems with a θ vacuum term. New ideas are clearly needed in orderto make significant advances in this problem, and one possibility is the development of newsimulation algorithms that might circumvent the difficulties of conventional approaches. Soour next logical step would be to combine the fermionic expansion of the MDP of Karsch andMutter with this strong coupling expansion for pure gauge fields, and try to build a completelynew algorithm, capable of dealing with fermions and gauge fields for µ = 0, and maybe, throughan intelligent clustering procedure, for µ 6= 0 as well.

9I must reckon that, some time before our development took place, S. Chandrasekharan had come across asimilar idea in [116], where he succeeded to simulate QED in four dimensions by using a worm algorithm and aworld-sheet approach. Indeed, the conclusions he reports in his papers are remarkably the same as ours.

135

136

Chapter 8

Further applications of polymers

“Defeat is not the worst of failures. Notto have tried is the true failure.”

—George Edward Woodberry

8.1 Mixing fermions and gauge theories: QED

The fermionic contributions of the MDP model share a common property: They all havetrivial contributions from the gauge links. In the case of dimers, the gauge link involved Un,µ is

annihilated by the reverse link Un+µ,−µ = U †n,µ; the baryonic loops consist of concatenated trivial

powers of the gauge links (in the case of SU(3), the third power U3n,µ); and the monomers have

no gauge links to play with. There are no closed loops of links contributing to the partitionfunction, even though some closed fermionic loops, giving non-vanishing contribution to theGrassman integral, carry a gauge loop associated. This happens because the group integral ofany closed, non-annihilating gauge loop is zero.

The addition of the strong coupling expansion of the gauge action changes this picture dra-matically. Now the non-annihilating gauge loops associated to fermionic loops can be cancelledby other gauge loops coming from the pure gauge action expansion. Fermions however, followFermi statistics and anticommutation rules, hence we expect a sign problem to appear, for thereare fermionic-gauge loops which carry a negative weight. The simplest of them all, for two di-mensions, is shown if fig. 8.1, but indeed there are many other examples. As we will see, thedynamical simulations of fermions and gauge fields with polymeric models are untractable atthis moment. In the following sections I will develop the model of the polymeric expansion ofQED with fermions, and show how the sign problem spoils the measurements of observables.

Pairing gauge and fermionic terms

As we previously solved the strong coupling expansion for abelian groups, we can afford toconstruct the partition function of the complete system, with fermions and gauge fields out ofthe strong coupling regime. The procedure to follow is the expansion of the exponentials whichdefine the partition function, in the same fashion we did in the previous chapters. To avoidrepetition, I just explain the differences. The whole partition function can be decomposed inthree terms

Z = ZMDP × ZGA + ZMixed (8.1)

137

X

X X

X

XX

X

XX

Figure 8.1: Simplest fermionic loop in the theory mixing fermions and gauge fields out of thestrong coupling regime. The solid line indicated a gauge loop, which annihilates the gaugecontribution of the fermionic loop, marked by a dashed line. The × mark lattice points, thisloop is larger than the simplest plaquette.

where ZGA is exactly given in (7.2), ZMDP comes from the U(1) version of (6.24) and ZMixed

aglomerates all the contributions which are not included in the previous partition functions.The new expression for ZMDP becomes quite simple in U(1) for two reasons: First, there areno baryons; and second, there are no colours, only one degree of freedom is allowed per site,and this degree of freedom can be either a dimer or a monomer

ZMDP =∑

α∈CF

CαF = (2m)Nα

M . (8.2)

where C denotes the whole space of configurations, and CαF is a factor which gathers the con-

tribution of monomers and dimers only. The terms encoded by ZMixed are those which involveproducts of gauge and fermionic loops. Now, a closed fermionic loop which does not annihilateall its gauge links can give rise to a contribution in the final partition function. For instance, aplaquettte loop,

ψnUn,µψn+µψn+µUn+µ,νψn+µ+νψn+µ+νU†n+ν,µψn+νψn+νU

†n,νψn

gives zero contribution to the MDP partition function because, even if the Grassman integralis non-zero, the gauge group integral of the plaquette defined by the fermionic line vanishes.The difference is that now, we can complete this term by multiplying it by the complementarygauge loop, coming from the gauge action,

Un,νUn+ν,µU †n+µ,νU

†n,µ.

This way all the Grassman variables and the gauge links are killed, and the integral over thegauge group becomes trivial again. As these fermionic loops consume all the anticommutingdegrees of freedom, they are self-avoiding, and cannot mix with monomers. The new rule forthe occupancy numbers of each site is

nM(n) +∑

ν

[nD(n, ν) + nF (n, ν)] = 1, (8.3)

which takes into account the fermionic lines nF which produce the fermionic loops. Since thereis only a pair ψ, ψ of Grassman variables for each site, the dimers become self-avoiding as well.This fact will prove a handicap in the clustering process.

The weight carried by these fermionic loops comprises a contribution coming from the gaugeplaquette, and a contribution coming from the closed fermionic line which defines the loop. Theformer can be absorbed in ZGA, whereas the latter is just a sign given by the Kogut-Susskindphases

138

w (Cα) =∏

{x,µ}∈CαLoops

η(x)µ (8.4)

where CαLoops is the set of fermionic loops belonging to the configuration Cα. Therefore, the

contribution of a general configuration involving monomers, dimers, gauge plaquettes and anti-plaquettes, and these new kind of fermionic loops is a product of the former contributions Cα

F

and CαG, multiplied by the fermionic loop weights

Z = (2m)NαM

∏

k

βnαk +nα

k

2nαk+nα

k nαk !nα

k !

∏

{x,µ}∈CαLoops

η(x)µ. (8.5)

As we can see, the sign associated to each fermionic loop gives rise to a potential sign problem.When close to the strong coupling limit, the sign problem becomes milder as expected, for thestrong coupling limit of this theory without baryons should be free of sign problems. But ifwe keep increasing the coupling β, at certain point the sign problem becomes severe. Thisbehaviour is related to the relative importance of each fermionic loop: At strong coupling, thereare no fermionic loops, as they involve gauge plaquettes, whose weight is proportional to apower of β. The configuration is basically a bunch of dimers and monomers, depending on themass of the fermion. As β increases, the simplest fermionic loops (plaquettes) appear, whichare positive. But at certain point, the negative loops (see fig. 8.1) become non-negligible, andthe sign problem becomes severe.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Mea

nSig

n〈σ〉

Beta β

× × ××

×

×

×

×× × × × × ××××××

×

×

×

×

××

Figure 8.2: Mean sign as a function of the coupling β for the Schwinger model with polymerizedKogut-Susskind fermions, keeping the ratio m/e fixed. We used a 1002 lattice, with 250000thermalization steps and 1000000 measurement steps. The sign problem seems severe beyondβ ∼ 1.1.

Some ideas for configuration clustering

The way Karsch and Mutter solved the sign problem for the MDP model at zero chemicalpotential should be an example of the power of the discrete models to deal with the signproblem. Configuration clustering could be the way out to solve this long-standing problem, asforetold long time ago [128]. As this model, involving gauge fields and fermions, is richer than

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the previous models analyzed in terms of objects which live in our lattice, we expect to findmore possibilities for the clustering.

Our clustering attempt gathered dimers, fermionic and gauge loops in the same term of thepartition function, that is, for each fermionic loop, a pair of dimers using the same numberof Grassman variables are associated to it, and a pair of gauge plaquette/anti-plaquette. Theaddition of such an object to an existing configuration would have a transition probabilityproportional to

Pi→j ∝β

2

[

σ +β

2

]

where sigma is the sign of the loop. For β ≥ 2, the sign problem could be overcome. Yet thisturned out to be an unnatural choice, for a set of dimers cannot be mirrored into fermionicloops in some cases. We need a one-to-one correspondence between fermionic loops, and dimersplus pairs of gauge plaquettes, but we can think of a very simple configuration of dimers andgauge plaquettes with no correspondence: A single dimer populating the lattice.

Figure 8.3: Attempt to cluster configurations in the new model. We add a gauge loop with twodimers (right, the horizontal thick solid lines represent the dimers) and a fermionic loop (left)into a single entity. Unfortunately, there is no one-to-one correspondence in this case, and anisolated single dimer has no correspondence with the proposed system.

In the end, we did not find a satisfactory way to cluster configurations and to solve the signproblem; however, this does not mean that the problem cannot be solved.

A word on Wilson fermions

The discussion has been restricted to Kogut-Susskind fermions up to now, but also a systemfeaturing Wilson fermions can be translated into a set of monomers, dimers and fermionic loops.The pioneer on this field is M. Salmhofer [118]1, which succeeded in solving the Schwinger modelat strong coupling with Wilson fermions. The procedure to follow in this taks is the same, butdue to the inclusion of the γ matrices, the Grassman rules which the points of the lattice mustcomply with are far more complex, and the number of possible points increase notoriously. Infact, M. Salmhofer mapped QED2 to an eight vertex model, where each vertex has a different(and positive) weigth, but the inclusion of gauge fields increases the number of vertex to 49[130]2, and a great number of these vertex are negative. The result is a sign problem, whichworsens as the dimension of the model increases, thus the models were dropped at some point,although it seems that no one ever tried to overcome the sign problem with the clusteringof configurations. This failure contrast with the success of other fermionic models, like theGross-Neveu model, or the Karsch and Mutter proposal for µ = 0.

1K. Scharnhost solved the two flavoured version in [129]. U. Wenger has recently worked on this topic as well[119], but it seems that no significant advances where made on his papers.

2It is remarkable the effort spend by the group of Graz –particularly C. Gattringer- in the implementation ofpolymeric models on the lattice. Some examples are colleted in [130, 131, 132].

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8.2 Other interesting models

The polymerization procedure can be applied to several interesting models, where it can be usedto try to solve the sign problem by clustering, or to reduce the critical slowing down. Indeed,taking into account the limited success of this kind of models and algorithms to reduce theseverity of the sign problem, the scientific community regard them as a way to avoid the criticalslowing down, as the authors of the worm algorithm explicitly state in [114]. The applicabilityof the method to statistical systems is quite wide.

The Ising model

I will briefly summarize the steps necessary to construct the equivalent polymeric model. Re-calling (5.3), we apply the identities

eFsisj = cosh F [1 + sisj tanh F ] ,

ehsi = cosh h [1 + si tanhh] ,

to Z. The new partition function is written as

Z (F, h) = (coshF )N∑

{si}

∏

{i,j}(1 + sisj tanhF )

×

(cosh h)N

[∏

i

(1 + si tanh h)

]

. (8.6)

There are two kinds of graphs derived from this expansion:

• Closed loops of dimers. The product of different sisj tanhF terms give non-zero con-tributions only when every spin is squared, for s2

i = 1. Otherwise, the sum over allconfigurations which involve the unpaired spin si = ±1 vanishes. In order to find anonzero contribution, we need to construct a chain like

si1si2 tanh F × si2si3 tanhF × si3si4 tanhF . . . sinsi1 tanhF,

where every spin appears twice. As sij and sij+1are neighbours (as indicated in the

product over {i, j} in (8.6)), this lead to a closed loop. The loops are not self-avoiding,and two loops may cross at a given lattice site, but overlapping dimers are not allowed,because we have used all the spin variables available. The weight of this kind of loops isgiven by the coupling F

wIsCl = (tanhF )L , (8.7)

with L the length of the loop. As L is always an even number, there is no sign problem,even at negative values of the coupling (the antiferromagnetic model).

• Open chains with two heads. The latter expansion need not give rise to a closed loop,if we cap both ends of the growing spin chain with monomers, coming from the externalfield term

si1 tanh hsi1si2 tanhF × si2si3 tanhF . . . sin−1sin tanhF × sin tanhh.

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The result is an open string, whose ends carry monomers (heads). The weight of thischain

wIsOp = (tanhF )L (tanhh)2 , (8.8)

can become negative, if the length of the string is odd and the coupling F is negative(antiferromagnetic coupling). Then a severe sign problem appears.

The model was tested successfully for the ferromagnetic case. A parallel work of U. Wolffwas published in [133], where the absence of critical slowing down is emphasized.

The Antiferromagnetic Ising model at θ = π

Although in general the Ising model under an imaginary magnetic field has a severe sign problem,this is a special case where simulations can be performed. The equivalence relations that areused here are

eFsjsk = cosh F + sjsk sinhF,

e−ı θ2sj = cos

θ

2− isj sin

θ

2. (8.9)

In general, the open chains in this model may have negative weights, but for θ = π the systemis greatly simplified. The former relations become

eFsjsk = cosh F + sjsk sinhF,

e−ıπ2

si = −isj . (8.10)

In the one-dimensional case, the chains of dimers are removed, no matter if they belong to aclosed or to an open loop, for the inner points of these chains are multiplied by −isj factors (persite j) which can not be annihilated, and therefore its weight in the partition function vanishesafter summing up all the possibilities for the spin. Thus, only dimers with their heads cappedwith monomers are the allowed objects of this theory, and their weight is simply

wD = sinhF. (8.11)

For higher dimensions, an odd number of dimers (up to 2D−1) can touch a site j: this structurebrings up an odd number of sj factors, that are cancelled by the −isj factor coming from theexternal field. Thus complex structures like nets can be constructed. The system of dimers canbe summed up analytically for one and two dimensions [94, 95], and can be simulated for higherdimensions easily. This special case, which seems pretty accidental, give us however a strongreason to keep believing in polymeric models.

The Hubbard model

The Hubbard model is a simple model, describing tighly coupled charged fermions (electronsor holes) on a crystal. It is quite interesting due to its relationship to superconductivity: Itwas thought than the main properties of the superconductors at high temperature should bedescribed by this model [134, 135].

The one-dimensional model can be analytically solved, and shows no superconducting phase.The two-dimensional model was expected to behave better, with supercoducting behaviour whenthe chemical potential surpassed some critical value [136].

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We were interested in the two-dimensional repulsive Hubbard model with chemical potential,for it features a sign problem. The lattice formulation of the model was taken from [137],developed by M. Creutz. The quantum Hamiltonian of the Hubbard model, written in terms ofcreation and annihilation operators, is transformed into a lattice system of Grassman variables,after a discretization of the time direction. The original Hubbard Hamiltonian follows:

H = −K∑

{i,j},σa†iσajσ − U

2

∑

i

(

a†i↑ai↑ − a†i↓ai↓)2

+

h∑

i

(

a†i↑ai↑ − a†i↓ai↓)

+ µ∑

i,σ

a†iσaiσ, (8.12)

where K is the hopping parameter, U is the repulsion term (attractive if U < 0) betweenfermions, h is the magnetic field affecting the electrons according to their spin, µ the chemicalpotential which introduces an asymmetry in the spin distribution, and a†iσ and aiσ are thecreation and annihilation operators, verifying the anticommutation relation

{

aiσa†jρ

}

= δijδσρ. (8.13)

After performing the transformations described in [137], which involve the addition of a Gaussianscalar field, the final result for the partition function is

Z = limK

LT→0

eV 2β(U2

+h−µ)

(2π)V 2LT

2

∫

(dA) e−A2

2 e−ψ∗M+ψe−ψ∗M−ψ, (8.14)

with

ψ∗M±ψ = limK

LT→0

Kβ

LT

∑

{i,j},tψ∗

i,tψj,t −∑

i,t

ψ∗i,tψi,t−1+

∑

i,t

ψ∗i,tψi,te

“

βULT

”

12 Ai,t− β

LT(h+U±µ)

. (8.15)

Take into account that, in this model, no explicit regularization of fermions is used, only a setof Grassman variables to reproduce Fermi statistic.

The bosonic field is integrated out exactly, and the result is absorbed in the value of theweights for the monomers. After integrating the Grassman variables, we are left with four kindsof objects

• Monomers, ultralocal objects living in a site, whose weight depends on the monomeroccupation number nM of that site and of the spin of the monomer as

wD ∝

eσ2

−Λpm nM = 1

e2σ2

−Λ+−Λ− nM = 2

(8.16)

where σ and Λ are two constants

σ2 =βU

LT, Λpm =

β (U + h ± µ)

LT,

depending on the parameters of the theory.

143

• Spatial loops, extending only in the spatial plane, carrying a negative weight. This breakscompletely the possibility of simulating the model in the computer, for the number ofloops in a given configuration is compeltely arbitrary, and a configuration contributesnegatively to the partition function if it has an odd number of spatial loops. The weightof a loop is given by

wSL ∝ −(

βK

LT

)C

, (8.17)

with C the length of the loop.

• Temporal loops, winding around the lattice k times. These loops need not be straight,and can turn around in the spatial directions, but must always move forward in the timedirection. Due to the antiperiodic boundary conditions required for the time direction,the sign associated to the weight of these loops is (−1)k+1. The weight of these loopsdepends on the displacements on the spatial direction as

wTL ∝ (−1)k+1

(βK

LT

)D

, (8.18)

where D is the number of displacements in the spatial directions. In order to close theloop, D must always be an even number. As the only loop which is not displaced is thepositive weighted, straight loop with k = 1, this is the predominant loop, and loops ofhigher winding numbers are suppresed with powers of K/LT , which should vanish in thethermodynamic limit.

The behaviour of this model, regarding the mean sign of a simulation, is quite similar to thatof the MDP model at µ 6= 0: The sign problem is a real problem, as long as the configurationsare far from saturation of loops (in this case, straight temporal loops). In a large lattice, thissign problem always appears. We did not come across a clusterization of configurations whichcould solve this sign problem, even at zero chemical potential, where the standard theory doesnot suffer from a sign problem.

This case is an example of complete failure of the polymerization procedure: It takes awell-behaved model (the 2 + 1 Hubbard model at zero chemical potential), and transforms itinto a model with a severe sign problem. What we can learn from this story is the fact that, asa rule, a polymeric version of a given fermionic or spin theory does not solve the sign problem,and even might worsen it. On the other hand, for some systems this procedure might giveunvaluable help, speeding up simulations, reducing the critical slowing down, or even allowingus to perform simulations in a system with the sign problem.

8.3 Conclusions

The polymerization procedure arised originally as a method to simulate fermions in the earlydays of the lattice. Although it allowed to perform simulations of fermions at a low computercost, it introduced a sign problem, severe in most cases. Nonetheless, the polymerized systemsare discrete, and this property enables us to play with the clustering of configurations. Ithas been demonstrated that a clever clustering can solve completely the sign problem. Sadly,the addition of a chemical potential usually breaks the sign equilibrium of the configurationclusters, bringing back the sign problem. That is why the MDP model works so fine at zerochemical potential, but when µ 6= 0, no matter what we do, the sign problem comes back for low

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temperatures and large volumes. In fact, the configurations unbalanced by a chemical potentialare very hard to cluster into useful groups. Therefore, and unless a lattice researcher comes witha brilliant idea, the polymerization of fermions do not seem to be the solution to the problemof chemical potential, even though there is still some hope lying in the utilisation of specifictechniques for discrete systems, like the merom algorithm.

This conclusion takes us back to the original purpose of the polymers: to simulate fermionsat a very low cost. Gauged models in the strong coupling regime seem to work fine, and pureabelian3 theories also are tractable [139]. But when one mixes fermions and gauge fields, thesign problem is back, and the clustering becomes difficult. Nevertheless, this kind of modelshave a great advantage with respect to the non-zero chemical potential case: the configurationsare not strongly unbalanced by the addition of e±µ factors to some configurations. This way, Ifind quite plausible to think that a sensible way to cluster configurations exists, like the one thatKarsch and Mutter found in the MDP model. In addition, from time to time this proceduresurprises us finding a model with a sign problem, which can be completely solved after thepolymerization, like the n−dimensional Ising model at θ = π.

On the whole, the polymerization is a technique very easy to apply which might prove tobe very useful in some systems, and that should not be discarded at first glance, but certainlyit will not solve the sign problem.

3Non-abelian pure gauge theories seem to have a sign problem [138].

145

146

Summary

In this work, several long-standing problems of QCD have been analysed from the lattice pointof view. Although the main motivation driving this research was the sign problem, the varietyof topics treated here is more extense, covering the realization of symmetries in QCD andalternative methods to simulate fermions on the computer. Here an overview of the conclusionsdrawn in this dissertation is exposed, classified by chapter number. Since the first two chaptersare introductory, they lack conclusions, so our enumeration will begin from chapter three:

Chapter 3: The Aoki phase

Up to now, all the dynamical fermionic simulations of lattice QCD, done inside the Aokiphase with the aim of investigating the spontaneous breaking of Parity and Flavour, used atwisted mass term hiψγ5τ3ψ, which broke explicitly both symmetries [38, 39, 41, 42]. The zeroexternal field limit h → 0 was taken afterwards, in order to see if the breaking of the symmetriesstill remained in the absence of the source. In principle, this method (the addition and removalof an external source) is a valid method to investigate SSB, mostly in analytical calculations;nonetheless it has some handicaps when we put it in the computer:

• Since the thermodynamic V → ∞ and the zero external source h → 0 limits do notcommute, a great number of simulations must be performed in order to reach the ther-modynamic limit for each value of the external source, and at the end an uncontrolledextrapolation to zero external field must be made. These two facts introduce large sys-tematic errors, quite difficult to handle.

• On the other hand, the addition of an external source like hiψγ5τ3ψ enforces the standardAoki vacuum, verifying

⟨iψγ5ψ

⟩= 0,

⟨iψγ5τ3ψ

⟩6= 0.

• Moreover, not all the external sources are allowed with this method; those introducinga sign problem can not be simulated. This was an obstacle to prove our claims of newphases, for we could not select one of the new vacua we claim to have discovered.

The p.d.f. formalism allows us to study the vacuum structure of the Aoki phase withoutadding external sources to the action, thus it circumvents all the aforementioned dangers andproblems of the external source method. However, the p.d.f. requires previous knowledge ofthe spectrum of the hermitian Dirac operator γ5D, therefore several possibilities appeared,depending on the behaviour of the eigenvalues µi of γ5D:

(i.) If the spectral density of γ5D of each configuration U , ρU (µ, κ) is asymmetric in thethermodynamic limit, and the symmetry is only recovered in the continuum limit a → 0,then the expectation value

147

⟨(iψγ5ψ

)2⟩

may become negative. This fact would complicate the η meson measurements in the physi-cal phase (outside Aoki’s), unless the measurements are done when close to the continuumlimit, where the spectral symmetry is expected to be recovered. Moreover, assuming thatthe Aoki phase disappears at some finite value of the coupling β, as some authors suggest[45, 58], the physical interpretation of the Aoki phase could be compromised, for if the

expected value of the square of the hermitian pseudoscalar operator⟨(

iψγ5ψ)2

⟩

can be

negative, then the correlation function that defines the η meson mass may cast incon-gruous results. This possibility was discarded after performing quenched simulations oflattice QCD, inside and outside the Aoki phase, for several volumes (44, 64 y 84).

(ii.) The second possibility assumes that the spectral symmetry is recovered in the thermody-namic limit. Then, the standard Aoki phase,

⟨iψγ5ψ

⟩= 0,

⟨iψγ5τ3ψ

⟩6= 0;

is not the only possibility, but new vacua appear, characterized by

⟨iψγ5ψ

⟩6= 0.

These vacua are not connected to Aoki’s original one by Flavour or Parity transformations.The result is quite unexpected, for both, Aoki’s approximate calculations [27, 28] andχPT [32, 33], support unambiguously

⟨iψγ5ψ

⟩= 0, in spite of the spontaneous breaking

of Parity. The reason explaining why⟨iψγ5ψ

⟩vanishes in a vacuum with broken Parity

is the existence of a symmetry called P ′, which is a composition of Parity and a discreteFlavour rotation.

The existence of that symmetry can be called into question thanks to some inconsistencies.The first one is related to the fact that both, Parity and Flavour, are spontaneouslybroken, therefore, we expect any symmetry, composition of Parity and Flavour, to bebroken as well. This argument is not very strong, but is reasonable. On the other hand,the non-local gluonic operator

W =1

V

N∑

i

1

µi

is an order parameter of the P ′ symmetry. Were W intensive, 〈Wn〉 should vanish inthe thermodynamic limit for any natural n in a P ′ conserving scenario. Nevertheless, W

contributes to the expectation value⟨(

iψγ5ψ)2

⟩

in such a way that, if⟨W 2

⟩vanishes,

⟨(iψγ5ψ

)2⟩

must forcibly become non-zero inside the Aoki phase. The operator W scales

with the volume in the same way as an intensive operator, thus this is an strong argumentsupporting the existence of new phases.

In any case, the p.d.f. calculations are exact, hence (i.) either the calculations of Aoki,Sharpe and Singleton use approximations which are not accurate enough to predict theexistence of new phases, or (ii.) there must exist forcibly a way to reconcile the p.d.f.results with these calculations. In fact, an attempt has been proposed in [43]; in this papera solution involving the realization of an infinite set of sum rules for the eigenvalues of the

148

γ5D is introduced, however there are no theoretical arguments supporting the realizationof these sum rules.

In order to find which of the two possibilities (sum rules or new phases) actually happens inQCD simulations, we performed dynamical fermionic simulations inside the Aoki phasewithout external sources. From the technical point of view, this simulations are quitechallenging for the Dirac operator develops small eigenvalues inside the Aoki phase, itscondition number grows with V , and becomes hard to invert. That is why a new algorithm,using recent techniques [52], was specailly develop to address this problem. The numerical

results are not conclusive: although the expectation value⟨(

iψγ5ψ)2

⟩

is clearly non-zero

in the smallest volume 44, the measurements for a higher volume 64 are noisy enough toprevent us from obtaining a final result.

149

Chapter 4: Parity conservation in QCD from first principles

In spite of the tacit assumption that QCD does not break spontaneously neither Paritynor any global vector symmetry, there is a notorius lack of proofs of this hypothesis. Thewell-known theorems of Vafa and Witten [59, 60] –in particular, the one dedicated to Parity[59]– have been called into question several times in the literature [32, 61, 62, 63, 65, 66], andnowadays everybody agrees on the lack of a proof of Parity conservation in QCD. Regarding thevector-symmetry conservation theorem [60], it requires such demanding conditions, that onlythe naive and the Kogut-Susskind regularizations of fermions on the lattice comply with them.Moreover, the theorem is not applicable neither to Wilson fermions (one of the most widespreadregularizations in QCD simulations), nor to Ginsparg-Wilson fermions. Remarkably, the Wilsonfermions violate both theorems, breaking Parity and Flavour spontaneously in the so-called Aokiphase.

The probability distribution function formalism p.d.f., combined with an appropiate regu-larisation, is capable of proving Parity and Flavour conservation (and in general. if the p.d.f.is correctly used, the conservation of any other vector symmetry) in QCD. By appropiate onemust understand that in this regularisation, the eigenvalues of the Dirac operator have a lowerbound for a non-zero value of the quark mass. Wilson fermions violate this condition, and thesmall eigenvalues of the Dirac operator are responsible for the breaking of Parity and Flavourin the Aoki phase. Nevertheless, the Ginsparg-Wilson fermions have good chiral properties, sothe masses are multiplicatively renormalized. Then the eigenvalues of the Dirac operator arebounded from below for massive quarks, and the conservation of Parity and Flavour in QCD canbe proved within the p.d.f. framework, overcoming the difficulties found in the earlier attempsof Vafa and Witten.

Chapter 5: The antiferromagnetic Ising model within an imaginary magnetic field

Even though the final aim of the study of the antiferromagnetic Ising model within animaginary magnetic field is to test techniques and algorithms to simulate the more relevant caseof QCD with a θ term, the results obtained for the Ising model deserve attention on their own.By using the method described in section 5.3, the order parameter of the Z2 symmetry canbe computed for any value of θ in the ordered phase. In spite of the failure of the method todeliver sensible results for the low-coupling region, the data obtained allow us to make reasonableassumptions on the phase diagram of the theory, assumptions that were corroborated in a mean-field calculation.

If one adds to this information the existing results for the antiferromagnetic Ising modelat θ = π [94, 95], the phase diagram of the two-dimensional model can be reconstructed qual-itatively, matching that of the mean-field theory, and although higher dimensions were notpursued in this work, we expose some convincing arguments, which strongly suggest that thisphase diagram holds for higher dimensions.

Sadly the method employed here to solve the model is not exempt from flaws: firts, it doesnot work properly if there is a phase transition for θ < π; in this case it could give wrongresults. Lastly, the extrapolations required by the method are only reliable for high values fothe coupling |F |. Fortunately for us, a transition in θ < π is not expected in QCD, therefore thefirst point should not be relevant for our future work. Regarding the second point, QCD is anasymptotically free theory, thus its continuum limit lies in the region were the extrapolations ofthe method work well. That is why this method profiles itself as the perfect candidate to explorewith the computer QCD with a θ term, possibly a necessary deed in order to understand thetopological properties of QCD.

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Chapter 6: QCD and the chemical potential µ

The MDP model, developed by Karsch and Mutter [109], successfully solved the sign prob-lem of the polymeric formulation of QCD at µ = 0 for non-vanishing masses. The model couldbe easily modified to admit non-zero values for the chemical potential, so the extension wascompulsory. However, the original MDP implementation of Karsch and Mutter displayed er-godicity problems for small values of the quark mass, and the dynamics were ruled mainly bytwo states of the system: a saturated state, were all the space points are taken up by a baryonicloop, resulting in maximum baryonic density, and a depleted state with no baryonic loops at all.As these two states carry a positive weight, the sign problem aparently dissappeared, thanks tothe ergodicity problems of the original implementation.

Nevertheless, once the ergodicity problem is solved, the system is allowed to be in much morestates, which could carry a positive or a negative weight; then, the sign problem comes back. Inorder to see these effects, we need a large spatial volume, otherwise the system interacts withthe walls or with itself (depending on the boundary conditions), resulting in a saturation stateright after the transition. The observed saturation seems not to be a physical property of thesystem4 [101, 102, 103]; in fact it can be solved by increasing the spatial volume of the system.For larger volumes, a core of baryonic loops can be observed in the dense phase (for µ ∼> µc),whereas the borders are empty. It is this emptiness that allows for the creation and annihilationof loops on the surface of the baryonic core; as for high values of µ the loops carrying positiveand negative weights are almost equally important in the partition function, both kind of loopsare expected to appear on the core surface, creating large sign fluctuations. If, on the contrary,the spatial volume is small, the core takes up all the available space, saturation occurrs, andthe sign problem disappears, but this is a finite volume effect. Indeed, the sign problem seemsunavoidable, for it is created by the Fermi statistics, and Fermi statistics rule the dynamics ofthe dense phase.

The failure of the MDP model is a heavy blow for all the finite density QCD practitioners.Nonetheless, there still remain unexplored possibilities, applicable to discrete systems as theMDP model, that might mitigate or even solve the sign problem. That is why it is a sensibleidea to keep researching in those kind of models.

Chapter 7 and 8: Algorithms for the pure gauge action and Further applications of polymers

The polymerization of the fermionic action appeared as a method to simulate fermions onthe lattice. Even if this approach allowed us to simulate fermions at a low computationalcost, it introduced a (severe in most cases) sign problem. However, the polymeric systems arediscrete systems as well, and this property proved quite useful when it comes to clusteringconfigurations with opposite signs, in order to cancel the sign fluctuations by hand. It wasdemonstrated [109] that if one performs a clever clustering in some polymeric models, thesevere sign problem disappears completely. But the addition of a chemical potential to theaction breaks the fragile equilibrium between positive and negative weighted configurations,bringing the sign problem back to the game. This is the case of the MDP model, which worksproperly (after a clever clustering of configurations) at zero chemical potential, but for µ 6= 0the sign problem returns and spoils the behaviour of the system. Indeed, a set of configurationswhich has been unbalanced by a chemical potential is quite difficult to cluster in such a waythat the sign problem disappears. That is why, unless a new brilliant idea go on stage, thepolymerization of fermions seems not to offer a solution to the problem of finite density QCD,although it is true that not every possibility of this technique has been explored yet.

4Although saturation may occurr for very high values of µ.

151

This conclusion takes us back to the original purpose of the polymers: to simulate fermionson the lattice at low computational cost. The pure gauge models in the strong coupling regimecan be simulated with polymers, even those that feature baryonic loops, thanks to the contribu-tion of Karsch and Mutter [109]. It happens that the abelian gauge theories are polymerizable5

without introducing a sign problem [139]. But when one mixes polymerized fermions with poly-merized gauge fields, the sign problem comes back, and a new clustering that might solve thesign problem seems difficult to find. Nevertheless, this case has an important advantage withrespect to the chemical potential case: the configurations are not strongly unbalanced by theexponential e±µ factors. This way it seems plausible that a sensible way to cluster configura-tions should exist, following the steps of Karsch and Mutter, who found a way in the MDPmodel. In addition, the polymerization gives us some surprises from time to time: some systemsfeaturing a severe sign problem can be completely solved or simulated by using this technique,like, for instance, the Ising model at θ = π.

On the whole, the polymerization technique can be applied easily to any system, and al-though it does not represent the definitive solution for the sign problem, it can prove useful forsome particular systems, and should not be discarded a priori.

5The polymerization of non-abelian gauge theories introduce a severe sign problem [138].

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Outlook

Although in some particular cases we have achieved a remakable success, most of the problemsof QCD treated here remain unsolved after our analysis. Nonetheless this contribution shouldbe very useful to anybody interested in these topics. In the following lines I explain whatremains unsolved, and which are going to be our future lines of investigation.

• The structure of the Aoki phase remains unclear. Neither us nor the advocates ofthe standard picture of the Aoki phase have provided enough arguments to clarify thisissue. Future research is needed to find out what happens in this region.

• Parity and Flavour have been proved from first principles in this work, by a clever useof the p.d.f. formalism. This topic is closed.

• Regarding QCD with a θ term, the methods used in chapter five seem quite promising,and will be applied to QCD with the aim of finding more about the topology of QCD,and maybe, about the strong CP problem.

• The problem of the chemical potential in QCD can not be solved using polymers.Another approach must be taken.

• On the other hand, the polymers might prove useful to analyze other systems in-volving fermions, spins or gauge fields. We have not explored thoroughly the possibilityof simulating non-abelian gauge fields, for instance. Many trends of investigation are stillopen, and shall be pursued.

153

154

Appendix A

Nother’s theorem

In classical field theories, the existence of continuous symmetries is associated to conservedquantities. This relationship was established long time ago by Emmy Nother [140]. Her theoremstates:

”For any differentiable symmetry of the action of a physical system, there corresponds aconservation law”

In general, given a field φ(x), one can perform a coordinate transformation, involving only thecoordinates xµ, an internal transformation involving only the field φ, or a general transformationwhich combines both. It is convenient to compute the total variation of a local function underan infinitesimal general transformation:

φ(x) → φ′(x′)

∆φ = φ′(x′) − φ(x) = φ′(x + δx) − φ(x) ∼=∼= φ′(x) − φ(x) + δxµφ,µ = δφ

︸︷︷︸

Internal

transformation

+ δxµφ,µ︸ ︷︷ ︸

Coordinate

transformation

, (A.1)

were we are using the standard notation φ,µ = ∂µφ.

Let’s consider a general lagrangian density L, which depends on the coordinatex xµ, somefields and their first derivatives. We concentrate on the effects of transforming a single field φand computing the variation of the corresponding action

δS =

∫

d4x′L(φ′, ∂′µφ′, x′) −

∫

d4xL(φ, ∂µφ, x).

As we are dealing with an infinitesimal transformation, we can write x′ ∼= x + δx, so the newjacobian becomes

∂x′

∂x= 1 + ∂µδxµ,

and then expand L around L. For the shake of clarity, we will call L(φ′, ∂′µφ′, x′) just L and

L(φ, ∂µφ, x) simply L. Then,

∫

d4x′L =

∫

d4x (1 + ∂µδxµ)L (φ(x) + δφ, φ,µ + δφ,µ, x + δx) ∼=

155

∼=∫

d4x

[

L +∂L∂φ

δφ +∂L∂φ,µ

δφ,µ +∂L∂xµ

δxµ

]

+

∫

d4xL∂µδxµ.

The expression above can be simplified enormously. First of all, we integrate by parts the lastterm:

∫

d4x∂µδxµL =

∫

d4x∂µ (L∂xµ) −∫

d4x∂L∂xµ

δxµ,

which leaves us with∫

d4x′L ∼=∫

d4x

[

L +∂L∂φ

δφ +∂L∂φ,µ

δφ,µ

]

+

∫

d4x∂µ (L∂xµ) .

After this, we use the following property of our continuous infinitesimal transformation

δφ,µ = ∂µδφ,

which allows us to integrate by parts the last term inside the brackets,∫

d4x∂L∂φ,µ

δφ,µ =

∫

d4x∂µ

(

δφ∂L∂φ,µ

)

−∫

d4x∂µ

(∂L∂φ,µ

)

δφ,µ.

Combining both, we compute the total variation of the action

δS ∼=∫

d4x

[∂L∂φ

− ∂µ

(∂L∂φ,µ

)]

δφ +

∫

d4x∂µ

(

L∂xµ + δφ∂L∂φ,µ

)

.

The term in brackets vanishes by virtue of the equations of motion. Thence, if our transforma-tion is a symmetry of the action, δS = 0, and the following equation holds

∫

d4x∂µ

(

L∂xµ + δφ∂L∂φ,µ

)

= 0.

Recalling (A.1), the case of a general transformation can be explicitly solved

δφ → ∆φ = δφ + δxµ ∂φ

∂xµ

∫

d4x∂µ

(

L∂xµ + δφ∂L∂φ,µ

+ δxµφ,µ∂L∂φ,µ

)

= 0,

but in general we are not interested in coordinate frame changes, and our concerns sway aroundthe internal transformations. Therefore we set δxµ to zero to find

∫

d4x∂µ

(

δφ∂L∂φ,µ

)

= 0. (A.2)

As this result must hold for any value of δφ, the integral is irrelevant here, and we define thecurrent associated to our internal symmetry jµ as

jµ = δφ∂L∂φ,µ

. (A.3)

Equation (A.2) immediately imposes∂µjµ = 0,

and the current is conserved. Integrating j0 in a time-slice we find the conserved charge asso-ciated to the current

Q =

∫

d3xj0.

156

Appendix B

Hubbard-Stratonovich identity andSaddle-Point Equations

B.1 The Hubbard-Stratonovich identity

The Hubbard-Stratonovich identity is used to linearize quadratic exponents. The derivation isquite simple: Let’s begin from the equality

π12 =

∫ ∞

−∞e−x2

dx. (B.1)

The definite integral on the r.h.s. of (B.1) is invariant under translations, due to the infinitelimits. Thence

π12 =

∫ ∞

−∞e−(x−a)2dx =

∫ ∞

−∞e−x2+2axe−a2

dx. (B.2)

The factor e−a2

is a constant, and we can move it out of the integral to the l.h.s.

ea2

=1

π12

∫ ∞

−∞e−x2+2axdx. (B.3)

By substituting a → ia we can deal with negative exponents

e−a2

=1

π12

∫ ∞

−∞e−x2+2iaxdx. (B.4)

In the infinite ranged Ising model, a =(

FN

) 12∑N

i si, and the quadratic term in the expo-nential is substituted by an integral in a ghost variable x

eFN (

PNi si)

2

=1

π12

∫ ∞

−∞e−x2+2x( F

N )12

PNi sidx. (B.5)

B.2 The Saddle-Point Equations

Let f (x) be a continuous function whose absolute maximum lies at x0. Let ZN be

ZN =

∫ ∞

−∞eNf(x)dx. (B.6)

Then

157

limN→∞

1

NlnZN = f (x0) . (B.7)

Proof:

First we take out of the exponential a common factor

ZN =

∫ ∞

−∞eNf(x)dx = eNf(x0)

∫ ∞

−∞e−N(f(x0)−f(x))

︸ ︷︷ ︸

IN

dx, (B.8)

so as to make the integrand ≤ 1 for any value of x. Now we compute

limN→∞

1

NlnZN = f (x0) + lim

N→∞1

Nln [IN ] . (B.9)

Since ZN is finite for finite N , the integral IN is also finite for a finite value of N . Moreover,as N increases, the value of the integral should stay or decrease. Therefore

∀N > N0 IN0≥ IN =⇒ 1

Nln IN ≤ 1

Nln IN0

,

and as N → ∞, the quantity 1N ln IN0

vanishes, thus we impose the condition

limN→∞

1

Nln IN ≤ 0. (B.10)

This completes the first part of the proof.

Now we take advantage of the continuity of f (x):

∀ε ∃δ > 0‖ |x − x0| < δ =⇒ |f (x) − f (x0)| < ε.

Given a number ε, I find a δ satisfying the continuity condition, and divide the integral IN inthree parts,

IN =

∫ ∞

−∞e−N(f(x0)−f(x)) =

∫ x0−δ

−∞e−N(f(x0)−f(x))dx+

∫ x0+δ

x0−δe−N(f(x0)−f(x))dx +

∫ ∞

x0+δe−N(f(x0)−f(x))dx. (B.11)

As all these integrals are positive, the following inequality,

IN ≥∫ x0+δ

x0−δe−N(f(x0)−f(x))dx, (B.12)

is straightforward, but f (x0) − f (x) is bounded by ε, so

IN ≥∫ x0+δ

x0−δe−Nεdx = e−Nε2δ. (B.13)

Taking logarithms and dividing by N

1

NIN ≥ 1

N(ln 2 + ln δ) − ε. (B.14)

The first term of the r.h.s. dies in the limit N → ∞. The second term can be arbitrarily small,thence

158

limN→∞

1

Nln IN ≥ 0. (B.15)

which is the second part of the proof. Combined with the first piece

0 ≥ 1

Nln IN ≥ 0, (B.16)

and we can rewrite equation (B.9) taking into account (B.16),

limN→∞

1

NlnZN = f (x0) + lim

N→∞1

Nln [IN ] = f (x0) , (B.17)

which is the desired result.

159

160

Appendix C

Mean-field theory

Let us solve the infinite coupled, ferromagnetic Ising model, described by the Hamiltonian

H (J, B, {si}) = − J

N

N∑

i6=j

sisj − BN∑

i

si. (C.1)

Defining F = βJ and h = βB, the partition function

Z (F, h) =∑

{si}e

FN

PNi6=j sisj+h

PNi si (C.2)

can be summed up by noticing the following equality

N∑

i6=j

sisj =

[N∑

i

si

]2

− N (C.3)

and it is this equality that allow us to recover (5.61) up to a constant. Now the quadraticexponent is linearized using the Hubbard-Stratonovich identity1

Z (F, h) =e−F

π12

∫ ∞

−∞

∑

{si}e−x2+

„

2x( FN )

12 +h

«

PNi si

dx. (C.4)

The integrand factorizes, as there is no spin-spin interaction

Z (F, h) =2Ne−F

π12

∫ ∞

−∞e−x2

coshN

[

2x

(F

N

) 12

+ h

]

dx. (C.5)

Let us remove N factors by rescaling the variable x

x → N12 y

dx → N12 dy

so (C.4) becomes

Z (F, h) = 2Ne−F

(N

π

) 12∫ ∞

−∞

[

e−y2+ln

h

cosh“

2F12 y+h

”i]N

dy. (C.6)

1See Appendix B

161

Again, the integral (C.6) is not solvable by standard means, but the saddle-point technique2

enables us to work out the free energy

limN→∞

1

NlnZ (F, h) = ln 2+

limN→∞

1

Nln

∫ ∞

−∞

[

e−y2 ln

h

cosh“

2F12 y+h

”i]N

dy. (C.7)

And the saddle-point equations

−y0 + F12 tanh

(

2F12 y0 + h

)

= 0, (C.8)

−1 +2F

cosh2(

2F12 y0 + h

) < 0, (C.9)

maximize the following function

g (y) = −y2 + ln[

cosh(

2F12 y + h

)]

. (C.10)

Thus, the free energy is

f (F, h) = ln 2 + g (y0) (C.11)

and y0 complies with the saddle-point equations (C.9). It is straightforward to link the ghostvariable y to the magnetization of the system

〈m〉 = m0 =∂f

∂h=

∂g

∂h

∣∣∣∣y=y0

+

∂g

∂y

∣∣∣∣y=y0

∂y

∂h= tanh

(

2F12 y0 + h

)

= F− 12 y (C.12)

This way we arrive to the standard mean-field equation3

m0 = tanh (2Fm0 + h) . (C.13)

The dependency of the free energy on the external field is hidden in m0 (h).Equation (C.13) is the standard mean-field equation. Indeed, when performing mean field

calculations, people usually try an ansatz like (C.13), instead of deriving it, as it has been donein this work, and this ansatz normally guesses right.

The ansatz method is based on physical grounds, and begins with equation (5.58) for para-magnetic substances

m = tanh (h) . (C.14)

This equation only takes into account the external magnetic field, neglecting the magnetic fieldthat might appear if the magnetic dipoles of the substance align to h. This induced field iscalled molecular field, and we expect it to be proportional to the current magnetization of thesample m. As magnetic interactions are quite short ranged, we expect each dipole to interact

2See Appendix B.3In some text, a 2D factor appears accounting for the coordination number, leaving the equation as m0 =

tanh (4DFm0 + h). This factor can be absorbed in the coupling constant F , and does not modify qualitativelythe physics of the model.

162

only with surrounding dipoles, so the strength of the molecular field is also proportional to thecoordination number q and to a coupling constant dipole-magnetization F . Therefore, our guessis

m = tanh (qFm + h) . (C.15)

which is essentially the same as (C.13).These equations are valid for the ferromagnetic case, however, the antiferromangetic case

is not so straightforward to guess. A simple but effective way would be to substitute themagnetization m by the staggered magnetization mS in (C.15). But this must be wrong, forthe magnetic field affects the spins in the same way as the staggered magnetization, even thoughthe external field is not staggered itself.

There are other approaches, like the one showed in [96], where the lattice is divided in twodifferent sublattices S1 and S2, and the following mean-field equations are proposed

m1 = 12 tanh (2Fm2 + h) , (C.16)

m2 = 12 tanh (2Fm1 + h) . (C.17)

These equations are suspicious in the sense that, for a ferromagnetic coupling F > 0, they failto deliver the original equation (C.13) in the limit m1 = m2 = m

2 , giving instead

m = tanh (Fm + h) . (C.18)

On the contrary, our derivation

m1 = 12 tanh (h + 2FmS) , (C.19)

m2 = 12 tanh (h − 2FmS) . (C.20)

leads to the right result if we tweak the couplings a bit. We divide FmS into

m1 = 12 tanh (2 (F11m1 + F12m2) + h) , (C.21)

m2 = 12 tanh (2 (F21m1 + F22m2) + h) . (C.22)

where Fij represents the coupling of spins belonging to the sublattice Si to the ones belonging tothe sublattice Sj . The ferromagnetic case is represented by F11 = F22 = F12 = F21 > 0, whereasthe antiferromagnetic case appears when F11 = F22 = −F12 = −F21 > 0. The main differencebetween the two approaches is the fact that we have taken into account all the spins whencomputing the molecular field, whereas our colleagues in [96] took only into account the spinsof the opposite sublattice. Indeed, if in our original result for the antiferromagnetic model weintroduce a staggered reduced magnetic field hS , which enhances the staggered magnetization4

instead of a standard field h, we find for mS an equation which is analogous to the one for min the ferromagnetic case

mS = tanh (2FmS + hS) . (C.23)

This is an expected result, an indicates that we are moving along the right track, for the modelsare completely equivalent.

4This effect can be achieved by allowing the field to couple differently to the two sublattices.

163

Coming back to the ferromagnetic model, equation (C.13) displays spontaneous magneti-zation for certain values of the coupling F . The best way to check this is by solving equation(C.13) graphically, as it is done in fig. C. Fig. C tells us which solutions of (C.13) are stable(maxima) and which are not (minima).

-1.00

-0.50

0.00

0.50

1.00

m− m+-1.00 -0.50 0.00 0.50 1.00

Mag

net

izat

ion

m

Magnetization m

Figure C.1: Graphical solution to equation (C.13). The existence of spontaneous symmetrybreaking depends on the slope of the function tanh (2Fm0) at the origin, as discussed in thischapter.

The parameter who determines the right number of solutions to the saddle-point equationis F , through the first derivative of the function tanh (2Fm0) at the origin

∂

∂m0tanh (2Fm0)

∣∣∣∣m0=0

= 2F. (C.24)

Depending on the behaviour close to the origin, we have several possibilities

• If the slope of tanh (2Fm0) is higher than the slope of the function y = m0

2F > 1 → F >1

2,

there are three solutions to the saddle point equation, m0 = {m+, m−, 0}, two of themsymmetric to keep the original Z2 symmetry of the action.

• On the contrary, a small slope of tanh (2Fm0), F < 12 , leads to only one solution at the

origin, adn the Z2 symmetry is preserved.

The value FC = 12 is called the critical coupling, and marks a second order phase transition with

diverging susceptibility

limF→F−

C

χ|m0=0 = limF→F−

C

∂m0

∂h

∣∣∣∣m0=0

=1 − m2

0

(1 − 2F )(1 − m2

0

) =⇒ χ → ∞, (C.25)

and the critical exponent γ for this divergence is γ = 1. It can be proved that the exponentremains the same if we approach the critical point from above (F → F+

C ).Finally, the effect of the reduced external field on the broken phase is to select a vacuum

from the two distinct possibilities.

164

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

m− m+-1.00 -0.50 0.00 0.50 1.00

Fre

eE

ner

gyf

Magnetization m

Figure C.2: Free energy (C.11) as a function of the magnetization m0. Whenever spontaneoussymmetry breaking occurrs, the stable vacua are those far from the origin, which violate Z2

symmetry. For the non-SSB scenario, the only stable vacua is at the origin, and the meanmagnetization is zero. Whether the first or the second condition happens depends only on F .

165

166

Appendix D

Some mean-field results for the Isingmodel

In order to compute the gap in the susceptibilities at the critical line, it is compulsory to findout the behaviour of the variable y0 in the neighbourhood of θc. What we must do is solve thesaddle-point equation

y0 =|F |

12

2

sinh(

4 |F |12 y0

)

cosh2(

2 |F |12 y0

)

− sin2 θ2

. (D.1)

The way to proceed is to expand the hyperbolic functions as a power series in y, and drop allthe higher order terms. As for θ < θc the only solution to the saddle point equation is y0 = 0,we can expand around this point

sinh(

4 |F |12 y0

)

= 4 |F |12 y0 +

32

3|F |

32 y3

0 + O(y5

), (D.2)

cosh(

2 |F |12 y0

)

= 1 + 2 |F | y20 + O

(y4

), (D.3)

so the saddle-point equation becomes

y0 ∼ 2 |F | y0 + 83 |F | y3

0

4 |F | y20 + cos2 θ

2

y0 << 1. (D.4)

Now we expand again the denominator of (D.4) up to y20,

1

4 |F | y20 + cos2 θ

2

=1

cos2 θ2

− 4 |F |cos4 θ

2

y20 + O

(y40

). (D.5)

Therefore, for θ ∼> θc,

y0 ∼ 2 |F |cos2 θ

2

y0 + 8 |F |2 2 cos2 θ2 − 3

3 cos4 θ2

y30. (D.6)

We already know of the y0 = 0 solution. Solving the quadratic equation that is left,

y0 =

√

3

8 |F |2

(cos2 θ

2 − 2 |F |)cos2 θ

2

2 cos2 θ2 − 3

. (D.7)

167

Thus y0 tends to zero as θ approaches the critical value for a given F . Its derivative with respectto θ, on the other hand,

dy0

dθ= −

√

3

32 |F |2sin θ

(cos4 θ

2 − 3 cos2 θ2 + 3 |F |

)

√(2 cos2 θ

2 − 3)3 [(

cos2 θ2 − 2 |F |

)cos2 θ

2

], (D.8)

diverges as

1√

(cos2 θ

2 − 2 |F |)cos2 θ

2

,

for at the critical line cos θc

2 = 2F . The divergence cancels in the product y0dy0

dθ . As y0 =√

|F |mS , this also applies to mSdmS

dθ .The solution obtained in (D.7) can be used to calculate the behaviour of the susceptibilities

around the critical point. The ‘topological’ susceptibility

χT =dm

di θ2

=dm

dθ

dθ

di θ2

=

(cos2 θ

2 − 1)cosh (2 |F |mS) + 1 − cos2 θ

2(cosh2 (2 |F |mS (θ)) − sin2 θ

2

)2 −

−2 |F | sin θ sinh (4 |F |mS) dmS

dθ(cosh2 (2 |F |mS) − sin2 θ

2

)2 , (D.9)

takes the value

limθ→θ−c

χT =1

cos2 θc

2

=1

2 |F | , (D.10)

as we approach θc from below. However, if we come from the antiferromagnetic phase θ > θc,the second term gives a non-zero contribution, for the derivative dmS

dθ diverges at the criticalline. The divergence is cancelled exactly by the factor sinh (4 |F |mS), as explained before, andwhat remains is a finite contribution

mSdmS

dθ

∣∣∣∣θ=θc

= − 3 sin θc

16 |F |2 (4 |F | − 3)

2 |F | sin θ sinh (4 |F |mS) dmS

dθ(cosh2 (2 |F |mS) − sin2 θ

2

)2 ∼ − 3 sin2 θc

32 |F |2 (4F − 3)

In the end

limθ→θ+

c

χT =1

2 |F | +3

4 |F |2 |F | − 1

4 |F | − 3, (D.11)

and the gap is

∆χT = limθ→θ+

c

χT − limθ→θ−c

χT =3

4 |F |2 |F | − 1

4 |F | − 3. (D.12)

The staggered susceptibility diverges at the critical line. This is quite expected, as forθ = 0 the susceptibility diverges at the critical point. The computation for any value of θ iscomplicated, for in order to obtain χS , we need to take derivatives with respect to a staggeredfield θS , and the take the θS → 0 limit. To this purpose, we use the original form of the freeenergy (5.67) with a θS term

168

f (F, mS , θ, θS) = − |F |m2S +

1

2ln

[

cosh

(

2 |F |mS +iθ + iθS

2

)

×

cosh

(

2 |F |mS − iθ − iθS

2

)]

(D.13)

Taking derivatives with respect to mS we should recover the saddle-point equation with theaddition of the θS source

df

dmS= 0 = −2 |F |mS + |F |

[

tanh

(

2 |F |mS +iθ + iθS

2

)

+

tanh

(

2 |F |mS − iθ − iθS

2

)]

. (D.14)

This time the saddle point equation is complex, therefore a working solution is not guaranteedat first glance. Nevertheless, we are using θS as a mathematical tool to find out the staggeredsusceptibility. In the end the θS → 0 limit is taken, and the validity of the saddle-point solutionis recovered.

From (D.14) a new equation for the staggered magnetization is obtained

mS =1

2

[

tanh

(

2 |F |mS +iθ + iθS

2

)

+

tanh

(

2 |F |mS − iθ − iθS

2

)]

. (D.15)

The derivative with respect to θS gives us the longed susceptibility

χS =dmS

d iθS

2

=1 + 2 |F |χS

2 cosh2(

2 |F |mS + iθ+iθS

2

)+

1 + 2 |F |χS

2 cosh2(

2 |F |mS − iθ−iθS

2

) =1 + 2 |F |χS

2X, (D.16)

where

X =1

2 cosh2(

2 |F |mS + iθ+iθS

2

) +1

2 cosh2(

2 |F |mS − iθ−iθS

2

) .

Moving all the terms proportional to χS to the l.h.s.

2χS (1 − |F |X) = X, (D.17)

we can find the value of χS

χS =X

2 − 2 |F |X . (D.18)

The quantity X must be evaluated at the point θ = θc and θS = 0. This is not a difficult taskand the final value is

X =1

|F | .

Therefore

169

χS =1

2 |F | − 2 |F | = ∞, (D.19)

and the susceptibility diverges at the critical line.Finally, and to elucidate the behaviour of m (θ) as θ → π, we need to work out the following

limit

limθ→π

dmS

dθsin θ. (D.20)

As sin θ → 0 when θ approaches π, only if the derivative dmS

dθ diverges at θ = π is the product(D.20) non-vanishing. The expansion we performed previously is not very useful here, as thepoint θ = π is far from the critical line (unless we are taking the F → 0 limit as well). Theway to solve this problem is to compute implicitly the derivative from the saddle-point equation(5.83) at θ = π

dmS

dθ

∣∣∣∣θ=π

=dmS

dθ

2 |F | cosh (4 |F |mS)

cosh2 (2 |F |mS) − sin2 θ2

∣∣∣∣∣θ=π

−

− sinh (4 |F |mS)dmS

dθ |F | sinh (4 |F |mS) − sin θ4

(cosh2 (2 |F |mS) − sin2 θ

2

)2

∣∣∣∣∣θ=π

=

=dmS

dθ

∣∣∣∣θ=π

4 |F | cotanh (2 |F |mS) [1 − cotanh (2 |F |mS)] . (D.21)

Moving all the terms to the l.h.s. of the equation we find that either

1 − 4 |F | cotanh (2 |F |mS) [1 − cotanh (2 |F |mS)] = 0, (D.22)

ordmS

dθ

∣∣∣∣θ=π

= 0. (D.23)

The first case is impossible, for the solution to the saddle-point equation at θ = π imposes

mS (π) = cotanh (2 |F |mS) , (D.24)

which is non-zero and verifies|mS | ≥ 1,

so the l.h.s. never vanishes, for the second summand is always positive. Therefore, (D.23)applies and the derivative vanishes at θ = π.

170

Appendix E

Spectrum of the Ginsparg-Wilsonoperator on the lattice

Years ago Ginsparg and Wilson (Ginsparg-Wilson) [11] suggested, in order to avoid the Nielsenand Ninomiya non go theorem [8] and to preserve chiral symmetry on the lattice, to require thefollowing condition for the inverse Dirac operator

γ5D−1 + D−1γ5 = 2aRγ5, (E.1)

where a is the lattice spacing and R is a local operator. Accordingly D should satisfy, insteadof the standard anticommutation relation of the continuum formulation, the Ginsparg-Wilsonrelation

γ5D + Dγ5 = 2aDRγ5D. (E.2)

Fifteen years after this proposal, Hasenfratz [141] and Neuberger [142] found that the fixed pointaction for QCD and the overlap fermions satisfy respectively the Ginsparg-Wilson relation, thelast with R = 1/2. Furthermore, Hasenfratz, Laliena and Niedermayer [143] realized thatGinsparg-Wilson fermions have nice chiral properties, allowing us to establish an exact indextheorem on the lattice. Indeed if we define a local density of topological charge as

q(x) = aRTr (γ5D (x, x)) , (E.3)

the corresponding topological charge is

Q = aRTr(γ5D), (E.4)

which is a topological invariant integer that approaches the continuum topological charge in thecontinuum limit. Finally by replacing the Ginsparg-Wilson Dirac operator D + m by

∆ + m =(

1 − am

2

)

D + m, (E.5)

in order to define an unsubtracted proper order parameter [144]

ψ

(

1 − aD

2

)

ψ, (E.6)

then the Ginsparg-Wilson fermionic action possesses an exact symmetry which is anomalousfor the flavour singlet transformations, but exact for the flavour non-singlet case (see [145]); aproperty which allows us to introduce also a θ parameter in the Ginsparg-Wilson action, as inthe continuum.

171

In this dissertation, the value R = 1/2 is used, as well as the massive Dirac operator (E.5)associated to the unsubtracted chiral order parameter (4.39), but the choice is irrelevant for thefinal results, as they hold as well for the standard, substracted order parameters, and for anyother constant values of R > 0.

Let us start with the Ginsparg-Wilson relation for R = 1/2,

γ5D + Dγ5 = aDγ5D. (E.7)

One also chooses D such that

γ5Dγ5 = D†. (E.8)

Merging (E.7) and (E.8),

D + D† = aDD† = aD†D. (E.9)

Hence D is a normal operator, and as such, it has a basis of orthonormal eigenvectors. Alsoeigenvectors corresponding to different eigenvalues are necessarily orthogonal. From (E.9) it isimmediate to check that the operator V = 1−aD is unitary, V †V = I. Therefore the spectrumof V lies in the unit circle with center in the origin, and the spectrum of D must then lie inthe shifted and rescaled circle of radius 1

a centered in the real axis at ( 1a , 0). Then, the possible

eigenvalues of D are of the form

λ =1

a

(1 − eiα

), α ∈ R, (E.10)

and the following identity is satisfied

λ + λ∗ = aλλ∗. (E.11)

Let v be an eigenvector of D with eigenvalue λ, Dv = λv. Taking into account (E.7), arelationship between the eigenvalues of v and γ5v can be found,

Dγ5v = −γ5Dv + aDγ5Dv = −λ (γ5v + aDγ5v) , (E.12)

and using (E.11) we arrive to the final expression

D (γ5v) = − λ

1 − aλ(γ5v) = λ∗ (γ5v) . (E.13)

Thus, if v is an eigenvector of D with eigenvalue λ, then γ5v is another eigenvector witheigenvalue λ∗, and when λ is not real, then those two eigenvectors correspond to differenteigenvalues and must be orthogonal. On the other hand, restricting ourselves to the subspacecorresponding to real eigenvalues, λ = 0 or λ = 2

a , γ5 and D commute, and therefore we canfind a common basis of eigenvectors; in other words, we can find an orthonormal basis for whichthe eigenvectors of D corresponding to real eigenvalues are chiral. If we denote by n+ (n−) thenumber of eigenvectors of positive (negative) chirality in the subspace corresponding to λ = 0,and similarly n′+ (n′−) for the subspace corresponding to λ = 2

a , then the fact that Tr(γ5) = 0and Q = a

2Tr(γ5D) imply

n+ − n− = n′− − n′+, (E.14)

Q = n− − n+. (E.15)

172

Let us call V the size of the matrix D. Then the density of topological charge, defined asQV , is bounded in absolute value by 1,

∣∣∣QV

∣∣∣ ≤ 1.

The operator in the fermion action is

∆ + m =(

1 − am

2

)

D + m (E.16)

Its spectrum is trivially related to the spectrum of D; if λ are as before the eigenvalues of D,then the eigenvalues of (E.16) are

(1 − am

2

)λ + m. They still lie in a circle with the center in

the real axis, and the possible real eigenvalues are now m and 2a . We will always require that

0 < m < 2a , then the operator (E.16) preserves the position of the higher real eigenvalue [69].

We will also need the spectrum of H = γ5 (∆ + m). It is easy to see that H is an hermitianoperator, H† = H, and therefore has real spectrum µj . This spectrum can be worked out bynoting that the matrix γ5 (∆ + m) is block diagonal in the basis of eigenvectors of D. Callingvλ such and eigenvector with non-real eigenvalue λ, then

Hvλ = γ5

(1 − am

2

)λvλ + mvλ∗ =

(m + λ

(1 − am

2

))vλ∗ ,

Hvλ∗ = γ5

(1 − am

2

)λ∗vλ∗ + mvλ =

(m + λ∗ (

1 − am2

))vλ.

(E.17)

This is a block diagonal matrix, decomposable in 2 × 2 blocks like

(0 m + λ

(1 − am

2

)

m + λ∗ (1 − am

2

)0

)

. (E.18)

The diagonalization of this block yields a pair of real eigenvalues ±µ,

µ2 = m2 + λλ∗(

1 − am

2

)2+ m (λ + λ∗)

(

1 − am

2

)

(E.19)

For the case λ ∈ R, the results become quite different. Let vλ be an eigenvector of D ofchirality χ = ±1, that is, γ5vλ = χvλ. Then

Hvλ = γ5

(

1 − am

2

)

λvλ + mχvλ =(

m + λ(

1 − am

2

))

χvλ, (E.20)

and the expression for µ reads

µ =(

m + λ(

1 − am

2

))

χ. (E.21)

More explicitly, as λ ∈ R implies that either λ = 0 or λ = 2a , the only allowed values for µ are

µ = ±m for λ = 0, with degeneracy n+ and n−, and similarly µ = ±a2 with degeneracy n′+,

n′− when λ = 2a .

The above calculations imply a bound for µ at finite mass, as was remarked in [69],

µ2 ≥ m2, (E.22)

valid only if ma < 2, which is a reasonable requirement. Otherwise, the spectrum of theGinsparg-wilson operator breaks down.

The importance of the mass bound (E.22) can not be overemphasized. It is essential inthe proofs developed in this dissertation on symmetry conservations, and also rules out theexistence of an Aoki phase for Ginsparg-Wilson fermions. We might wonder how this happensin the Ginsparg-Wilson formulation, given the fact that the Ginsparg-Wilson operator doesnot anticommunte with γ5. The answer is related to equation (E.1): we can write a similarequation for Wilson fermions,

{γ5, ∆

−1}

= 2aRγ5, where R is a non-local operator, but in thecase of Ginsparg-Wilson fermions, this R turns out to be local. It is this locality that enforces

173

the depletion of the spectrum of small real eigenvalues, close to the origin, for the eigenvectorsof ∆ look like chiral solutions at long distances, and effectively it is as if the anticommutatorwas realized. So, no quasi-chiral, exceptional configurations are allowed, the Aoki phase iscompletely forbidden, and therefore, the symmetries are respected.

-Another interesting results are straightforward from this point on. For instance, we knowthat there exist an index theorem for Ginsparg-Wilson fermions [143], thus we can relate thezero modes of ∆ to the topological density and the topological susceptibility,

Let’s consider now the determinant of the operator

∆ + m +q

Vγ5

(

1 − aD

2

)

. (E.23)

Proceeding as before, another block-diagonal structure is found in the basis of eigenvectors ofD, and the contribution to the determinant coming from the block corresponding to a complexpair vλ, vλ∗ is given by

det

( (1 − am

2

)λ + m q

V

(1 − aλ

2

)

qV

(1 − aλ∗

2

) (1 − am

2

)λ∗ + m

)

=

m2 +

[

1 −(am

2

)2]

λλ∗ − q2

V 2

(

1 − a2λλ∗

4

)

, (E.24)

where we have used the identity (E.11). The determinant of this block gives the contributionof a pair of eigenvalues to the determinant

1 − αq2

V 2, (E.25)

with

α =1 − a2λλ∗

4

m2 +[

1 −(

am2

)2]

λλ∗. (E.26)

The minimum achievable value for the determinant of the block (E.24) is obtained by settingλ = 0. In that case, the product equals m2. As a result, the determinant is bounded from belowfor any non-zero value of the mass, and the following relationship applies

|α| ≤ 1

m2(E.27)

Regarding the chiral modes of chirality χ, they contribute with a factor

(

1 − am

2

)

λ + m +q

V

(

1 − aλ

2

)

χ (E.28)

For λ = 2a the final factor is just 2

a , whereas for a zero mode λ = 0 with chirality χ, theoutcoming value is m + χ q

V .

174

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